Introduction

It is quite often to see direct sum or direct product of groups, modules, vector spaces. Indeed, for modules over a ring $R$, direct products are also direct products of $R$-modules as well. On the other hand, the direct sum is a coproduct in the category of $R$-modules.

But what about tensor products? It is some different kind of product but how? Is it related to direct product? How do we write a tensor product down? We need to solve this question but it is not a good idea to dig into numeric works.

The category of bilinear or even $n$-multilinear maps

From now on, let $R$ be a commutative ring, and $M_1,\cdots,M_n$ are $R$-modules. Mainly we work on $M_1$ and $M_2$, i.e. $M_1 \times M_2$ and $M_1 \otimes M_2$. For $n$-multilinear one, simply replace $M_1\times M_2$ with $M_1 \times M_2 \times \cdots \times M_n$ and $M_1 \otimes M_2$ with $M_1 \otimes \cdots \otimes M_n$. The only difference is the change of symbols.

The bilinear maps of $M_1 \times M_2$ determines a category, say $BL(M_1 \times M_2)$ or we simply write $BL$. For an object $(f,E)$ in this category we have $f: M_1 \times M_2 \to E$ as a bilinear map and $E$ as a $R$-module of course. For two objects $(f,E)$ and $(g,F)$, we define the morphism between them as a linear function making the following diagram commutative: $\def\mor{\operatorname{Mor}}$

This indeed makes $BL$ a category. If we define the morphisms from $(f,E)$ to $(g,F)$ by $\mor(f,g)$ (for simplicity we omit $E$ and $F$ since they are already determined by $f$ and $g$) we see the composition

$$\mor(f,g) \times \mor(h,g) \to \mor(h,f)$$

satisfy all axioms for a category:

CAT 1 Two sets $\mor(f,g)$ and $\mor(f’,g’)$ are disjoint unless $f=f’$ and $g=g’$, in which case they are equal. If $g \neq g’$ but $f = f’$ for example, for any $h \in \mor(f,g)$, we have $g = h \circ f = h \circ f’ \neq g’$, hence $h \notin \mor(f,g)$. Other cases can be verified in the same fashion.

CAT 2 The existence of identity morphism. For any $(f,E) \in BL$, we simply take the identity map $i:E \to E$. For $h \in \mor(f,g)$, we see $g = h \circ f = h \circ i \circ f$. For $h’ \in \mor(g,f)$, we see $f = h’ \circ g = i \circ h’ \circ g$.

CAT 3 The law of composition is associative when defined.

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There we have a category. But what about the tensor product? It is defined to be initial (or universally repelling) object in this category. Let’s denote this object by $(\varphi,M_1 \otimes M_2)$.

For any $(f,E) \in BL$, we have a unique morphism (which is a module homomorphism as well) $h:(\varphi,M_1 \otimes M_2) \to (f,E)$. For $x \in M_1$ and $y \in M_2$, we write $\varphi(x,y)=x \otimes y$. We call the existence of $h$ the universal property of $(\varphi,M_1 \otimes M_2)$.

The tensor product is unique up to isomorphism. That is, if both $(f,E)$ and $(g,F)$ are tensor products, then $E \simeq F$ in the sense of module isomorphism. Indeed, let $h \in \mor(f,g)$ and $h’ \in \mor(g,h)$ be the unique morphisms respectively, we see $g = h \circ f$, $f = h’ \circ g$, and therefore

$$g = h \circ h' \circ g \\ f = h' \circ h \circ f$$

Hence $h \circ h’$ is the identity of $(g,F)$ and $h’ \circ h$ is the identity of $(f,E)$. This gives $E \simeq F$.

What do we get so far? For any modules that is connected to $M_1 \times M_2$ with a bilinear map, the tensor product $M_1 \oplus M_2$ of $M_1$ and $M_2$, is always able to be connected to that module with a unique module homomorphism. What if there are more than one tensor products? Never mind. All tensor products are isomorphic.

But wait, does this definition make sense? Does this product even exist? How can we study the tensor product of two modules if we cannot even write it down? So far we are only working on arrows, and we don’t know what is happening inside an module. It is not a good idea to waste our time on ‘nonsenses’. We can look into it in an natural way. Indeed, if we can find a module satisfying the property we want, then we are done, since this can represent the tensor product under any circumstances. Again, all tensor products of $M_1$ and $M_2$ are isomorphic.

A natural way to define the tensor product

Let $M$ be the free module generated by the set of all tuples $(x_1,x_2)$ where $x_1 \in M_1$ and $x_2 \in M_2$, and $N$ be the submodule generated by tuples of the following types:

$$(x_1+x_1',x_2)-(x_1,x_2)-(x_1',x_2) \\ (x_1,x_2+x_2')-(x_1,x_2)-(x_1,x_2') \\ (ax_1,x_2)-a(x_1,x_2) \\ (x_1,ax_2) - a(x_1,x_2)$$

First we have a inclusion map $\alpha=M_1 \times M_2 \to M$ and the canonical map $\pi:M \to M/N$. We claim that $(\pi \circ \alpha, M/N)$ is exactly what we want. But before that, we need to explain why we define such a $N$.

The reason is quite simple: We want to make sure that $\varphi=\pi \circ \alpha$ is bilinear. For example, we have $\varphi(x_1+x_1’,x_2)=\varphi(x_1,x_2)+\varphi(x_1’,x_2)$ due to our construction of $N$ (other relations follow in the same manner). This can be verified group-theoretically. Note

$$\varphi(x_1+x_1',x_2)=(x_1+x_1',x_2)+N \\ \varphi(x_1,x_2)+\varphi(x_1',x_2)=(x_1,x_2)+(x_1',x_2)+N$$

but

$$\varphi(x_1+x_1',x_2)-\varphi(x_1,x_2)-\varphi(x_1',x_2)=(x_1+x_1',x_2)-(x_1,x_2)-(x_1',x_2) +N = 0+N.$$

Hence we get the identity we want. For this reason we can write

$$\begin{aligned} (x_1+x_1')\otimes x_2 &= x_1 \otimes x_2 + x_1' \otimes x_2, \\ x_1 \otimes (x_2 + x_2') &= x_1 \otimes x_2 + x_1 \otimes x_2', \\ (ax_1) \otimes x_2 &= a(x_1 \otimes x_2), \\ x_1 \otimes (ax_2) &= a(x_1 \otimes x_2). \end{aligned}$$

Sometimes to avoid confusion people may also write $x_1 \otimes_R x_2$ if both $M_1$ and $M_2$ are $R$-modules. But before that we have to verify that this is indeed the tensor product. To verify this, all we need is the universal property of free modules.

By the universal property of $M$, for any $(f,E) \in BL$, we have a induced map $f_\ast$ making the diagram inside commutative. However, for elements in $N$, we see $f_\ast$ takes value $0$, since $f_\ast$ is a bilinear map already. We finish our work by taking $h[(x,y)+N] = f_\ast(x,y)$. This is the map induced by $f_\ast$, following the property of factor module.

Trivial tensor product

For coprime integers $m,n>1$, we have $\def\mb{\mathbb}$

$$\mb{Z}/m\mb{Z} \otimes \mb{Z}/n\mb{Z} = O$$

where $O$ means that the module only contains $0$ and $\mb{Z}/m\mb{Z}$ is considered as a module over $\mb{Z}$ for $m>1$. This suggests that, the tensor product of two modules is not necessarily ‘bigger’ than its components. Let’s see why this is trivial.

Note that for $x \in \mb{Z}/m\mb{Z}$ and $y \in \mb{Z}/n\mb{Z}$, we have

$$m(x \otimes y) = (mx) \otimes y = 0 \\ n(x \otimes y) = x \otimes(ny) = 0$$

since, for example, $mx = 0$ for $x \in \mb{Z}/m\mb{Z}$ and $\varphi(0,y)=0$. If you have trouble understanding why $\varphi(0,y)=0$, just note that the submodule $N$ in our construction contains elements generated by $(0x,y)-0(x,y)$ already.

By Bézout’s identity, for any $x \otimes y$, we see there are $a$ and $b$ such that $am+bn=1$, and therefore

$$\begin{aligned} x \otimes y &= (am+bn)(x \otimes y) \\ &=am(x \otimes y)+bn (x \otimes y) \\ &= 0. \end{aligned}$$

Hence the tensor product is trivial. This example gives us a lot of inspiration. For example, what if $m$ and $n$ are not necessarily coprime, say $\gcd(m,n)=d$? By Bézout’s identity still we have

$$d(x \otimes y) = (am+bn)(x \otimes y) = 0.$$

This inspires us to study the connection between $\mb{Z}/m\mb{Z} \otimes \mb{Z}/n\mb{Z}$ and $\mb{Z}/d\mb{Z}$. By the universal property, for the bilinear map $f:\mb{Z}/m\mb{Z} \times \mb{Z}/n\mb{Z} \to \mb{Z}/d\mb{Z}$ defined by

$$(a+m\mb{Z},b+n\mb{Z})\mapsto ab+d\mb{Z}$$

(there should be no difficulty to verify that $f$ is well-defined), there exists a unique morphism $h:\mb{Z}/m\mb{Z} \otimes \mb{Z}/n\mb{Z} \to \mb{Z}/d\mb{Z}$ such that

$$h \circ \varphi(a+m\mb{Z},b+n\mb{Z}) = h((a+m\mb{Z}) \otimes(b+n\mb{Z})) = ab+d\mb{Z}.$$

Next we show that it has a natural inverse defined by

$$\begin{aligned} g:\mb{Z}/d\mb{Z} &\to \mb{Z}/m\mb{Z} \otimes \mb{Z}/n\mb{Z} \\ a+d\mb{Z} &\mapsto (a+m\mb{Z}) \otimes (1+n\mb{Z}). \end{aligned}$$

Taking $a’ = a+kd$, we show that $g(a+d\mb{Z})=g(a’+\mb{Z})$, that is, we need to show that

$$(a+m\mb{Z})\otimes(1+n\mb{Z}) = (a'+m\mb{Z}) \otimes (1+n\mb{Z}).$$

By Bézout’s identity, there exists some $r,s$ such that $rm+sn=d$. Hence $a’ = a + ksn+krm$, which gives

$$\begin{aligned} (a'+m\mb{Z}) \otimes (1+n\mb{Z}) &= (a+ksn+krm+m\mb{Z}) \otimes(1+n\mb{Z}) \\ &= (a+ksn+m\mb{Z}) \otimes (1+n\mb{Z}) \\ &=(a+m\mb{Z}) \otimes(1+n\mb{Z}) + (ksn+m\mb{Z})\otimes(1+n\mb{Z}) \\ &=(a+m\mb{Z}) \otimes (1+n\mb{Z}) \end{aligned}$$

since

$$(ksn+m\mb{Z}) \otimes (1+n\mb{Z}) =n(ks+m\mb{Z}) \otimes (1+n\mb{Z}) = (ks+m\mb{Z}) \otimes(n+n\mb{Z}) = 0.$$

So $g$ is well-defined. Next we show that this is the inverse. Firstly

$$\begin{aligned} g \circ h((a+m\mb{Z}) \otimes(b+n\mb{Z})) &= g(ab+d\mb{Z})\\ &= (ab+m\mb{Z}) \otimes (1+n\mb{Z}) \\ &=b(a+m\mb{Z}) \otimes(1+n\mb{Z}) \\ &= (a+m\mb{Z}) \otimes (b+n\mb{Z}). \end{aligned}$$

Secondly,

$$\begin{aligned} h \circ g(a+d\mb{Z}) &= h((a+m\mb{Z}) \otimes(1+n\mb{Z})) \\ &= a+d\mb{Z}. \end{aligned}$$

Hence $g = h^{-1}$ and we can say

$$\mb{Z}/m\mb{Z} \otimes \mb{Z} /n\mb{Z} \simeq \mb{Z} /\gcd(m,n)\mb{Z}.$$

If $m,n$ are coprime, then $\gcd(m,n)=1$, hence $\mb{Z}/m\mb{Z} \otimes \mb{Z}/n\mb{Z} \simeq \mb{Z}/\mb{Z}$ is trivial. More interestingly, $\mb{Z}/m\mb{Z}\otimes \mb{Z}/m\mb{Z}=\mb{Z}/m\mb{Z}$. But this elegant identity raised other questions. First of all, $\gcd(m,n)=\gcd(n,m)$, which implies

$$\mb{Z}/m\mb{Z} \otimes \mb{Z}/n\mb{Z} \simeq \mb{Z}/\gcd(m,n)\mb{Z} \simeq \mb{Z}/\gcd(n,m)\mb{Z} \simeq\mb{Z}/n\mb{Z}\otimes\mb{Z}/m\mb{Z}.$$

Further, for $m,n,r >1$, we have $\gcd(\gcd(m,n),r)=\gcd(m,\gcd(n,r))=\gcd(m,n,r)$, which gives

$$(\mb{Z}/m\mb{Z}\otimes\mb{Z}/n\mb{Z})\otimes\mb{Z}/r\mb{Z} \simeq \mb{Z}/\gcd(m,n)\mb{Z}\otimes\mb{Z}/r\mb{Z} \simeq \mb{Z}/\gcd(m,n,r)\mb{Z} \\ \mb{Z}/m\mb{Z}\otimes(\mb{Z}/n\mb{Z} \otimes\mb{Z}/r\mb{Z}) \simeq \mb{Z}/m\mb{Z} \otimes\mb{Z}/\gcd(n,r)\mb{Z} \simeq \mb{Z}/\gcd(m,n,r)\mb{Z}$$

hence

$$(\mb{Z}/m\mb{Z}\otimes\mb{Z}/n\mb{Z})\otimes\mb{Z}/r\mb{Z} \simeq \mb{Z}/m\mb{Z}\otimes(\mb{Z}/n\mb{Z}\otimes\mb{Z}/r\mb{Z}).$$

Hence for modules of the form $\mb{Z}/m\mb{Z}$, we see the tensor product operation is associative and commutative up to isomorphism. Does this hold for all modules? The universal property answers this question affirmatively. From now on we will be keep using the universal property. Make sure that you have got the point already.

Tensor product as a binary operation

Let $M_1,M_2,M_3$ be $R$-modules, then there exists a unique isomorphism

$$\begin{aligned} (M_1 \otimes M_2) \otimes M_3 &\xrightarrow{\simeq} M_1 \otimes (M_2 \otimes M_3) \\ (x \otimes y) \otimes z &\mapsto x \otimes(y \otimes z) \end{aligned}$$

for $x \in M_1$, $y \in M_2$, $z \in M_3$.

Proof. Consider the map

$$\begin{aligned} \lambda_x:M_2 \times M_3 &\to (M_1 \otimes M_2)\otimes M_3 \\ (y,z) &\mapsto (x \otimes y ) \otimes z \end{aligned}$$

where $x \in M_1$. Since $(\cdot\otimes\cdot)$ is bilinear, we see $\lambda_x$ is bilinear for all $x \in M_1$. Hence by the universal property there exists a unique map of the tensor product:

$$\overline{\lambda}_x:M_2 \otimes M_3 \to (M_1 \otimes M_2) \otimes M_3.$$

Next we have the map

$$\begin{aligned} \mu_x: M_1 \times (M_2 \otimes M_3) &\to (M_1 \otimes M_2) \otimes M_3 \\ (x,y \otimes z) &\mapsto \overline{\lambda}_x(y \otimes z) \end{aligned}$$

which is bilinear as well. Again by the universal property we have a unique map

$$\overline{\mu}_x: M_1 \otimes (M_2 \otimes M_3) \to (M_1 \otimes M_2) \otimes M_3.$$

This is indeed the isomorphism we want. The reverse is obtained by reversing the process. For the bilinear map

$$\lambda_x':M_1 \times M_2 \to M_1 \otimes (M_2 \otimes M_3)$$

we get a unique map

$$\overline{\lambda'}_x: M_1 \otimes M_2 \to M_1 \otimes (M_2 \otimes M_3).$$

Then from the bilinear map

$$\mu'_x:(M_1 \otimes M_2) \times M_3 \to M_1 \otimes (M_2 \otimes M_3)$$

we get the unique map, which is actually the reverse of $\overline{\mu}_x$:

$$\overline{\mu'}_x:(M_1 \otimes M_2) \otimes M_3 \to M_1 \otimes (M_2 \otimes M_3).$$

Hence the two tensor products are isomorphic. $\square$

Let $M_1$ and $M_2$ be $R$-modules, then there exists a unique isomorphism

$$\begin{aligned} M_1 \otimes M_2 &\xrightarrow{\simeq} M_2 \otimes M_1 \\ x_1 \otimes x_2 &\mapsto x_2 \otimes x_1 \end{aligned}$$

where $x_1 \in M_1$ and $x_2 \in M_2$.

Proof. The map

$$\begin{aligned} \lambda:M_1 \times M_2 &\to M_2 \otimes M_1 \\ (x,y) &\mapsto y \otimes x \end{aligned}$$

is bilinear and gives us a unique map

$$\overline{\lambda}:M_1 \otimes M_2 \to M_2 \otimes M_1$$

given by $x \otimes y \mapsto y \otimes x$. Symmetrically, the map $\lambda’:M_2 \times M_1 \to M_1 \otimes M_2$ gives us a unique map

$$\overline{\lambda'}:M_2 \otimes M_1 \to M_1 \otimes M_2$$

which is the inverse of $\overline{\lambda}$. $\square$

Therefore, we may view the set of all $R$-modules as a commutative semigroup with the binary operation $\otimes$.

Maps between tensor products

Consider commutative diagram:

Where $f_i:M_i \to M_i’$ are some module-homomorphism. What do we want here? On the left hand, we see $f_1 \times f_2$ sends $(x_1,x_2)$ to $(f_1(x_1),f_2(x_2))$, which is quite natural. The question is, is there a natural map sending $x_1 \otimes x_2$ to $f_1(x_1) \otimes f_2(x_2)$? This is what we want from the right hand. We know $T(f_1 \times f_2)$ exists, since we have a bilinear map by $\mu = \varphi’ \circ (f_1\times f_2)$. So for $(x_1,x_2) \in M_1 \times M_2$, we have $T(f_1 \times f_2)(x_1 \otimes x_2) = \varphi’ \circ (f_1 \times f_2)(x_1,x_2) = f_1(x_1) \otimes f_2(x_2)$ as what we want.

But $T$ in this graph has more interesting properties. First of all, if $M_1 = M_1’$ an $M_2 = M_2’$, both $f_1$ and $f_2$ are identity maps, then we see $T(f_1 \times f_2)$ is the identity as well. Next, consider the following chain

$$\cdots \to M_1 \times M_2 \xrightarrow{(f_1 \times f_2)}M_1' \times M_2' \xrightarrow{(g_1 \times g_2)}M_1'' \times M_2''\to \cdots.$$

We can make it a double chain:

It is obvious that $(g_1 \circ f_1 \times g_2 \circ f_2)=(g_1 \times g_2) \circ (f_1 \times f_2)$, which also gives

$$T(g_1 \times g_2) \circ T(f_1 \times f_2) = T(g_1 \circ f_1 \times g_2 \circ f_2).$$

Hence we can say $T$ is functorial. Sometimes for simplicity we also write $T(f_1,f_2)$ or simply $f_1 \otimes f_2$, as it sends $x_1 \otimes x_2$ to $f_1(x_1) \otimes f_2(x_2)$. Indeed it can be viewed as a map

$$\begin{aligned} T:L(M_1, M_1') \times L(M_2,M_2') &\to L(M_1 \otimes M_2, M_1' \otimes M_2') \\ (f_1 \times f_2) &\mapsto f_1 \otimes f_2. \end{aligned}$$

Is perhaps the most important technical tools in commutative algebra. In this post we are covering definitions and simple properties. Also we restrict ourselves into ring theories and no further than that. Throughout, we let $A$ be a commutative ring. With extra effort we can also make it to non-commutative rings for some results but we are not doing that here.

In fact the construction of $\mathbb{Q}$ from $\mathbb{Z}$ has already been an example. For any $a \in \mathbb{Q}$, we have some $m,n \in \mathbb{Z}$ with $n \neq 0$ such that $a = \frac{m}{n}$. As a matter of notation we may also say an ordered pair $(m,n)$ determines $a$. Two ordered pairs $(m,n)$ and $(m’,n’)$ are equivalent if and only if

$$mn'-m'n=0.$$

But we are only using the ring structure of $\mathbb{Z}$. So it is natural to think whether it is possible to generalize this process to all rings. But we are also using the fact that $\mathbb{Z}$ is an entire ring (or alternatively integral domain, they mean the same thing). However there is a way to generalize it. $\def\mfk{\mathfrak}$

Multiplicatively closed subset

(Definition 1) A multiplicatively closed subset $S \subset A$ is a set that $1 \in S$ and if $x,y \in S$, then $xy \in S$.

For example, for $\mathbb{Z}$ we have a multiplicatively closed subset

$$\{1,2,4,8,\cdots\} \subset \mathbb{Z}.$$

We can also insert $0$ here but it may produce some bad result. If $S$ is also an ideal then we must have $S=A$ so this is not very interesting. However the complement is interesting.

(Proposition 1) Suppose $A$ is a commutative ring such that $1 \neq 0$. Let $S$ be a multiplicatively closed set that does not contain $0$. Let $\mfk{p}$ be the maximal element of ideals contained in $A \setminus S$, then $\mfk{p}$ is prime.

Proof. Recall that $\mfk{p}$ is prime if for any $x,y \in A$ such that $xy \in \mfk{p}$, we have $x \in \mfk{p}$ or $y \in \mfk{p}$. But now we fix $x,y \in \mfk{p}^c$. Note we have a strictly bigger ideal $\mfk{q}_1=\mfk{p}+Ax$. Since $\mfk{p}$ is maximal in the ideals contained in $A \setminus S$, we see

$$\mfk{q}_1 \cap S \neq \varnothing.$$

Therefore there exist some $a \in A$ and $p \in \mfk{p}$ such that

$$p+ax \in S.$$

Also, $\mfk{q}_2=\mfk{p}+Ay$ has nontrivial intersection with $S$ (due to the maximality of $\mfk{p}$), there exist some $a’ \in A$ and $p’ \in \mfk{p}$ such that

$$p' + a'y \in S.$$

Since $S$ is closed under multiplication, we have

$$(p+ax)(p'+a'y) = pp'+p'ax+pa'y+aa'xy \in S.$$

But since $\mfk{p}$ is an ideal, we see $pp’+p’ax+pa’y \in \mfk{p}$. Therefore we must have $xy \notin \mfk{p}$ since if not, $(p+ax)(p’+a’y) \in \mfk{p}$, which gives $\mfk{p} \cap S \neq \varnothing$, and this is impossible. $\square$

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As a corollary, for an ideal $\mfk{p} \subset A$, if $A \setminus \mfk{p}$ is multiplicatively closed, then $\mfk{p}$ is prime. Conversely, if we are given a prime ideal $\mfk{p}$, then we also get a multiplicatively closed subset.

(Proposition 2) If $\mfk{p}$ is a prime ideal of $A$, then $S = A \setminus \mfk{p}$ is multiplicatively closed.

Proof. First $1 \in S$ since $\mfk{p} \neq A$. On the other hand, if $x,y \in S$ we see $xy \in S$ since $\mfk{p}$ is prime. $\square$

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Ring of fractions of a ring

We define a equivalence relation on $A \times S$ as follows:

$$(a,s) \sim (b,t) \iff \exists u \in S, (at-bs)u=0.$$

(Proposition 3) $\sim$ is an equivalence relation.

Proof. Since $(as-as)1=0$ while $1 \in S$, we see $(a,s) \sim (a,s)$. For being symmetric, note that

$$(at-bs)u=0 \implies (bs-at)u=0 \implies (b,t) \sim (a,s).$$

Finally, to show that it is transitive, suppose $(a,s) \sim (b,t)$ and $(b,t) \sim (c,u)$. There exist $u,v \in S$ such that

$$(at-bs)v=(bu-ct)w=0.$$

This gives $bsv=atv$ and $buw = ctw$, which implies

$$bsvuw=atvuw=ctwsv \implies (au-cs)tvw =0.$$

But $tvw \in S$ since $t,v,w \in S$ and $S$ is multiplicatively closed. Hence

$$[(a,s) \sim (b,t)] \land [(b,t) \sim (c,u)] \implies (a,s) \sim (c,u).$$

$\square$

Let $a/s$ denote the equivalence class of $(a,s)$. Let $S^{-1}A$ denote the set of equivalence classes (it is not a good idea to write $A/S$ as it may coincide with the notation of factor group), and we put a ring structure on $S^{-1}A$ as follows:

$$(a/s)+(b/t)=(at+bs)/st, \\ (a/s)(b/t)=ab/st.$$

There is no difference between this one and the one in elementary algebra. But first of all we need to show that $S^{-1}A$ indeed form a ring.

(Proposition 4) The addition and multiplication are well defined. Further, $S^{-1}A$ is a commutative ring with identity.

Proof. Suppose $(a,s) \sim (a’,s’)$ and $(b,t) \sim (b’,t’)$ we need to show that

$$(a/s)+(b/t)=(a'/s')+(b'/t')$$

or

$$(at+bs)/st = (a't'+b's')/s't'.$$

There exists $u,v \in S$ such that

$$(as'-a's)u=0 \quad (bt'-b't)v=0.$$

If we multiply the first equation by $vtt’$ and second equation by $uss’$, we see

$$as'uvtt'-a'suvtt'+bt'vuss'-b'tvuss'=[(at)s't'+(bs)s't'-(a't')st-(b's')st]uv,$$

which is exactly what we want.

On the other hand, we need to show that

$$ab/st = a'b'/s't'.$$

That is,

$$\exists y \in S,(abs't'-a'b'st)y=0.$$

Again, we have

$$(as'-a's)u=(as'-a's)uvbt'=(abs't'-a'bst')uv=0, \\ (bt'-b't)v=(bt'-b't)vua's=(a'bst'-a'b'st)uv=0.$$

Hence

$$(abs't'-a'bst')uv+(a'bst'-a'b'st)uv=(abs't'-a'b'st)uv=0.$$

Since $uv \in S$, we are done.

Next we show that $S^{-1}A$ has a ring structure. If $0 \in S$, then $S^{-1}A$ contains exactly one element $0/1$ since in this case, all pairs are equivalent:

$$(at-bs)0=0.$$

We therefore only discuss the case when $0 \notin S$. First $0/1$ is the zero element with respect to addition since

$$0/1+a/s = (0s+1a)/1s = a/s.$$

On the other hand, we have the inverse $-a/s$:

$$-a/s+a/s = (-as+as)/ss=0/ss=0/1.$$

$1/1$ is the unit with respect to multiplication:

$$(1/1)(a/s)=1a/1s=a/s.$$

Multiplication is associative since

$$[(a/s)(b/t)](c/u)=(ab/st)(c/u)=abc/stu. \\ (a/s)[(b/t)(c/u)]=(a/s)(bc/tu)=abc/stu.$$

Multiplication is commutative since

$$ab/st+(-ba)/st=(abst-bast)/s^2t^2=0.$$

Finally distributivity.

$$(a/s+b/t)(c/u)=(c/u)(a/s+b/t)=[(at+bs)/st](c/u)=(act+bcs)/stu \\ (a/s)(c/u)+(b/t)(c/u)=ac/su+bc/tu=(actu+bcsu)/stu^2=(act/bcs)/stu$$

Note $ab/cb=a/c$ since $(abc-abc)1=0$. $\square$ $\def\mb{\mathbb}$

Cases and examples

First we consider the case when $A$ is entire. If $0 \in S$, then $S^{-1}A$ is trivial, which is not so interesting. However, provided that $0 \notin S$, we get some well-behaved result:

(Proposition 5) Let $A$ be an entire ring, and let $S$ be a multiplicatively closed subset of $A$ that does not contain $0$, then the natural map

$$\begin{aligned} \varphi_S: A &\to S^{-1}A \\ x &\mapsto x/1 \end{aligned}$$

is injective. Therefore it can be considered as a natural inclusion. Further, every element of $\varphi_S(S)$ is invertible.

Proof. Indeed, if $x/1=0/1$, then there exists $s \in S$ such that $xs=0$. Since $A$ is entire and $s \neq 0$, we see $x=0$, hence $\varphi_S$ is entire. For $s \in S$, we see $\varphi_S(s)=s/1$. However $(1/s)\varphi_S(s)=(1/s)(s/1)=s/s=1$. $\square$

Note since $A$ is entire we can also conclude that $S^{-1}A$ is entire. As a word of warning, the ring homomorphism $\varphi_S$ is not in general injective since, for example, when $0 \in S$, this map is the zero.

If we go further, making $S$ contain all non-zero element, we have:

(Proposition 6) If $A$ is entire and $S$ contains all non-zero elements of $A$, then $S^{-1}A$ is a field, called the quotient field or the field of fractions.

Proof. First we need to show that $S^{-1}A$ is entire. Suppose $(a/s)(b/t)=ab/st =0/1$ but $a/s \neq 0/1$, we see however

$$ab/st=0/1 \implies \exists u \in S, (ab-0)u=0 \implies ab=0.$$

Since $A$ is entire, $b$ has to be $0$, which implies $b/t=0/1$. Second, if $a/s \neq 0/1$, we see $a \neq 0$ and therefore is in $S$, hence we’ve found the inverse $(a/s)^{-1}=s/a$. $\square$

In this case we can identify $A$ as a subset of $S^{-1}A$ and write $a/s=s^{-1}a$.

Let $A$ be a commutative ring, an let $S$ be the set of invertible elements of $A$. If $u \in S$, then there exists some $v \in S$ such that $uv=1$. We see $1 \in S$ and if $a,b \in S$, we have $ab \in S$ since $ab$ has an inverse as well. This set is frequently denoted by $A^\ast$, and is called the group of invertible elements of $A$. For example for $\mb{Z}$ we see $\mb{Z}^\ast$ consists of $-1$ and $1$. If $A$ is a field, then $A^\ast$ is the multiplicative group of non-zero elements of $A$. For example $\mb{Q}^\ast$ is the set of all rational numbers without $0$. For $A^\ast$ we have

If $A$ is a field, then $(A^\ast)^{-1}A \simeq A$.

Proof. Define

$$\begin{aligned} \varphi_S:A &\to (A^\ast)^{-1}A \\ x &\mapsto x/1. \end{aligned}$$

Then as we have already shown, $\varphi_S$ is injective. Secondly we show that $\varphi_S$ is surjective. For any $a/s \in (A^\ast)^{-1}A$, we see $as^{-1}/1 = a/s$. Therefore $\varphi_S(as^{-1})=a/s$ as is shown. $\square$

---

Now let’s see a concrete example. If $A$ is entire, then the polynomial ring $A[X]$ is entire. If $K = S^{-1}A$ is the quotient field of $A$, we can denote the quotient field of $A[X]$ as $K(X)$. Elements in $K(X)$ can be naturally called rational polynomials, and can be written as $f(X)/g(X)$ where $f,g \in A[X]$. For $b \in K$, we say a rational function $f/g$ is defined at $b$ if $g(b) \neq 0$. Naturally this process can be generalized to polynomials of $n$ variables.

Local ring and localization

We say a commutative ring $A$ is local if it has a unique maximal ideal. Let $\mfk{p}$ be a prime ideal of $A$, and $S = A \setminus \mfk{p}$, then $A_{\mfk{p}}=S^{-1}A$ is called the local ring of $A$ at $\mfk{p}$. Alternatively, we say the process of passing from $A$ to $A_\mfk{p}$ is localization at $\mfk{p}$. You will see it makes sense to call it localization:

(Proposition 7) $A_\mfk{p}$ is local. Precisely, the unique maximal ideal is

$$I=\mfk{p}A_\mfk{p}=\{a/s:a \in \mfk{p},s \in S\}.$$

Note $I$ is indeed equal to $\mfk{p}A_\mfk{p}$.

Proof. First we show that $I$ is an ideal. For $b/t \in A_\mfk{p}$ and $a/s \in I$, we see

$$(b/t)(a/s)=ba/ts \in A_\mfk{p}$$

since $a \in \mfk{p}$ implies $ba \in \mfk{p}$. Next we show that $I$ is maximal, which is equivalent to show that $A_\mfk{p}/I$ is a field. For $b/t \notin I$, we have $b \in S$, hence it is legit to write $t/b$. This gives

$$(b/t+I)(t/b+I)=1/1+I.$$

Hence we have found the inverse.

Finally we show that $I$ is the unique maximal ideal. Let $J$ be another maximal ideal. Suppose $J \neq I$, then we can pick $m/n \in J \setminus I$. This gives $m \in S$ since if not $m \in \mfk{p}$ and then $m/n \in I$. But for $n/m \in A_\mfk{p}$ we have

$$(m/n)(n/m)=1/1 \in J.$$

This forces $J$ to be $A_\mfk{p}$ itself, contradicting the assumption that $J$ is a maximal ideal. Hence $I$ is unique. $\square$

Example

Let $p$ be a prime number, and we take $A=\mb{Z}$ and $\mfk{p}=p\mb{Z}$. We now try to determine what do $A_\mfk{p}$ and $\mfk{p}A_\mfk{p}$ look like. First $S = A \setminus \mfk{p}$ is the set of all entire numbers prime to $p$. Therefore $A_\mfk{p}$ can be considered as the ring of all rational numbers $m/n$ where $n$ is prime to $p$, and $\mfk{p}A_\mfk{p}$ can be considered as the set of all rational numbers $kp/n$ where $k \in \mb{Z}$ and $n$ is prime to $p$.

$\mb{Z}$ is the simplest example of ring and $p\mb{Z}$ is the simplest example of prime ideal. And $A_\mfk{p}$ in this case shows what does localization do: $A$ is ‘expanded’ with respect to $\mfk{p}$. Every member of $A_\mfk{p}$ is related to $\mfk{p}$, and the maximal ideal is determined by $\mfk{p}$.

---

Let $k$ be a infinite field. Let $A=k[x_1,\cdots,x_n]$ where $x_i$ are independent indeterminates, $\mfk{p}$ a prime ideal in $A$. Then $A_\mfk{p}$ is the ring of all rational functions $f/g$ where $g \notin \mfk{p}$. We have already defined rational functions. But we can go further and demonstrate the prototype of the local rings which arise in algebraic geometry. Let $V$ be the variety defined by $\mfk{p}$, that is,

$$V=\{x=(x_1,x_2,\cdots,x_n) \in k^n:\forall f \in \mfk{p}, f(x)=0\}.$$

Then what about $A_\mfk{p}$? We see since for $f/g \in A_\mfk{p}$ we have $g \notin \mfk{p}$, therefore for $g(x)$ is not equal to $0$ almost everywhere on $V$. That is, $A_\mfk{p}$ can be identified with the ring of all rational functions on $k^n$ which are defined at almost all points of $V$. We call this the local ring of $k^n$ along the variety $V$.

Universal property

Let $A$ be a ring and $S^{-1}A$ a ring of fractions, then we shall see that $\varphi_S:S \to S^{-1}A$ has a universal property.

(Proposition 8) Let $g:A \to B$ be a ring homomorphism such that $g(s)$ is invertible in $B$ for all $s \in S$, then there exists a unique homomorphism $h:S^{-1}A \to B$ such that $g = h \circ \varphi_S$.

Proof. For $a/s \in S^{-1}A$, define $h(a/s)=g(a)g(s)^{-1}$. It looks immediate but we shall show that this is what we are looking for and is unique.

Firstly we need to show that it is well defined. Suppose $a/s=a’/s’$, then there exists some $u \in S$ such that

$$(as'-a's)u=0.$$

Applying $g$ on both side yields

$$(g(a)g(s')-g(a')g(s))g(u)=0.$$

Since $g(x)$ is invertible for all $s \in S$, we therefore get

$$g(a)g(s)^{-1}=g(a')g(s')^{-1}.$$

It is a homomorphism since

$$\begin{aligned} h[(a/s)(a'/s')]&=g(a)g(a')g(s)^{-1}g(s')^{-1} \\ h(a/s)h(a'/s')&=g(a)g(s)^{-1}g(a')g(s')^{-1}, \end{aligned}$$

and

$$h(a/s+a'/s')=h((as'+a's)/ss')=g(as'+a's)g(ss')^{-1} \\ h(a/s)+h(a'/s')=g(a)g(s)^{-1}+g(a')g(s')^{-1}$$

they are equal since

$$\begin{aligned} g(as'+a's)g(ss')^{-1}&=g(as')g(ss')^{-1}+g(a's)g(ss')^{-1} \\ &=g(a)g(s')g(s)^{-1}g(s')^{-1}+g(a')g(s)g(s)^{-1}g(s')^{-1} \\ &=g(a)g(s)^{-1}+g(a')g(s')^{-1}. \end{aligned}$$

Next we show that $g=h \circ \varphi_S$. For $a \in A$, we have

$$h(\varphi_S(a))=h(a/1)=g(a)g(1)^{-1}=g(a).$$

Finally we show that $h$ is unique. Let $h’$ be a homomorphism satisfying the condition, then for $a \in A$ we have

$$h'(a/1)=h'(\varphi_S(a))=g(a).$$

For $s \in S$, we also have

$$h'(1/s)=h'((s/1)^{-1})=h'(\varphi_S(s)^{-1})=h'(\varphi_S(s))^{-1}=g(s)^{-1}.$$

Since $a/s = (a/1)(1/s)$ for all $a/s \in S^{-1}A$, we get

$$h'(a/s)=h'((a/1)(1/s))=g(a)g(s)^{-1}.$$

That is, $h’$ (or $h$) is totally determined by $g$. $\square$

---

Let’s restate it in the language of category theory (you can skip it if you have no idea what it is now). Let $\mfk{C}$ be the category whose objects are ring-homomorphisms

$$f:A \to B$$

such that $f(s)$ is invertible for all $s \in S$. Then according to proposition 5, $\varphi_S$ is an object of $\mfk{C}$. For two objects $f:A \to B$ and $f’:A \to B’$, a morphism $g \in \operatorname{Mor}(f,f’)$ is a homomorphism

$$g:B \to B'$$

such that $f’=g \circ f$. So here comes the question: what is the position of $\varphi_S$?

Let $\mfk{A}$ be a category. an object $P$ of $\mfk{A}$ is called universally attracting if there exists a unique morphism of each object of $\mfk{A}$ into $P$, an is called universally repelling if for every object of $\mfk{A}$ there exists a unique morphism of $P$ into this object. Therefore we have the answer for $\mfk{C}$.

(Proposition 9) $\varphi_S$ is a universally repelling object in $\mfk{C}$.

Principal and factorial ring

An ideal $\mfk{o} \in A$ is said to be principal if there exists some $a \in A$ such that $Aa = \mfk{o}$. For example for $\mb{Z}$, the ideal

$$\{\cdots,-2,0,2,4,\cdots\}$$

is principal and we may write $2\mb{Z}$. If every ideal of a commutative ring $A$ is principal, we say $A$ is principal. Further we say $A$ is a PID if $A$ is also an integral domain (entire). When it comes to ring of fractions, we also have the following proposition:

(Proposition 10) Let $A$ be a principal ring and $S$ a multiplicatively closed subset with $0 \notin S$, then $S^{-1}A$ is principal as well.

Proof. Let $I \subset S^{-1}A$ be an ideal. If $a \in S$ where $a/s \in I$, then we are done since then $(s/a)(a/s) = 1/1 \in I$, which implies $I=S^{-1}A$ itself, hence we shall assume $a \notin S$ for all $a/s \in I$. But for $a/s \in I$ we also have $(a/s)(s/1)=a/1 \in I$. Therefore $J=\varphi_S^{-1}(I)$ is not empty. $J$ is an ideal of $A$ since for $a \in A$ and $b \in J$, we have $\varphi_S(ab) =ab/1=(a/1)(b/1) \in I$ which implies $ab \in J$. But since $A$ is principal, there exists some $a$ such that $Aa = J$. We shall discuss the relation between $S^{-1}A(a/1)$ and $I$. For any $(c/u)(a/1)=ca/u \in S^{-1}A(a/1)$, clearly we have $ca/u \in I$, hence $S^{-1}A(a/1)\subset I$. On the other hand, for $c/u \in I$, we see $c/1=(c/u)(u/1) \in I$, hence $c \in J$, and there exists some $b \in A$ such that $c = ba$, which gives $c/u=ba/u=(b/u)(a/1) \in I$. Hence $I \subset S^{-1}A(a/1)$, and we have finally proved that $I = S^{-1}A(a/1)$. $\square$

As an immediate corollary, if $A_\mfk{p}$ is the localization of $A$ at $\mfk{p}$, and if $A$ is principal, then $A_\mfk{p}$ is principal as well. Next we go through another kind of rings. A ring is called factorial (or a unique factorization ring or UFD) if it is entire and if every non-zero element has a unique factorization into irreducible elements. An element $a \neq 0$ is called irreducible if it is not a unit and whenever $a=bc$, then either $b$ or $c$ is a unit. For all non-zero elements in a factorial ring, we have

$$a=u\prod_{i=1}^{r}p_i,$$

where $u$ is a unit) (invertible).

In fact, every PID is a UFD (proof here). Irreducible elements in a factorial ring is called prime elements or simply prime (take $\mathbb{Z}$ and prime numbers as an example). Indeed, if $A$ is a factorial ring and $p$ a prime element, then $Ap$ is a prime ideal. But we are more interested in the ring of fractions of a factorial ring.

(Proposition 11) Let $A$ be a factorial ring and $S$ a multiplicatively closed subset with $0 \notin S$, then $S^{-1}A$ is factorial.

Proof. Pick $a/s \in S^{-1}A$. Since $A$ is factorial, we have $a=up_1 \cdots p_k$ where $p_i$ are primes and $u$ is a unit. But we have no idea what are irreducible elements of $S^{-1}A$. Naturally our first attack is $p_i/1$. And we have no need to restrict ourselves to $p_i$, we should work on all primes of $A$. Suppose $p$ is a prime of $A$. If $p \in S$, then $p/1 \in S$ is a unit, not prime. If $Ap \cap S \neq \varnothing$, then $rp \in S$ for some $r \in A$. But then

$$(p/1)(r/rp)=1,$$

again $p/1$ is a unit, not prime. Finally if $Ap \cap S = \varnothing$, then $p/1$ is prime in $S^{-1}A$. For any

$$(a/s)(b/t)=ab/st=p/1,$$

we see $ab=stp \not\in S$. But this also gives $ab \in Ap$ which is a prime ideal, hence we can assume $a \in Ap$ and write $a=rp$ for some $r \in A$. With this expansion we get

$$ab=stp \implies rbp=stp \implies rb=st \implies (r/s)(b/t)=1/1.$$

Hence $b/t$ is a unit, $p/1$ is a prime.

Conversely, suppose $a/s$ is irreducible in $S^{-1}A$. Since $A$ is factorial, we may write $a=u\prod_{i}p_i$. $a$ cannot be an element of $S$ since $a/s$ is not a unit. We write

$$a/s=1/s[(u/1)(p_1/1)(p_2/1)\cdots(p_n/1)]$$

We see there is some $v \in A$ such that $uv=1$ and accordingly $(u/1)(v/1)=uv/1=1/1$, hence $u/1$ is a unit. We claim that there exist a unique $p_k$ such that $1 \leq k \leq n$ and $Ap \cap S = \varnothing$. If not exists, then all $p_j/1$ are units. If both $p_{k}$ and $p_{k’}$ satisfy the requirement and $p_k \neq p_k’$, then we can write $a/s$ as

$$a/s = \{1/s[(u/1)(p_1/1)\cdots(p_{k-1}/1)(p_{k+1}/1)\cdots(p_{k'-1}/1)(p_{k'+1}/1)\cdots(p_n/1)](p_k/1)\}(p_{k'}/1).$$

Neither the one in curly bracket nor $p_{k’}/1$ is unit, contradicting the fact that $a/s$ is irreducible. Next we show that $a/s=p_k/1$. For simplicity we write

$$b = u\prod_{i=1 \\ i \neq k}^{n} p_i, \quad a = bp_k.$$

Note $a/s = bp_k/s = (b/s)(p_k/1)$. Since $a/s$ is irreducible, $p_k/1$ is not a unit, we conclude that $b/s$ is a unit. We are done for the study of irreducible elements of $S^{-1}A$: it is of the form $p/1$ (up to a unit) where $p$ is prime in $A$ and $Ap \cap S = \varnothing$.

Now we are close to the fact that $S^{-1}A$ is also factorial. For any $a/s \in S^{-1}A$, we have an expansion

$$a/s=1/s[(u/1)(p_1/1)(p_2/1)\cdots(p_n/1)].$$

Let $p’_1,p’_2,\cdots,p’_j$ be those whose generated prime ideal has nontrivial intersection with $S$, then $p’_1/1, p’_2/1,\cdots,p’_j/1$ are units of $S^{-1}A$. Let $q_1,q_2,\cdots,q_k$ be other $p_i$’s, then $q_1/1,q_2/1,\cdots,q_k/1$ are irreducible in $S^{-1}A$. This gives

$$a/s = [(1/s)(p'_1/1)(p'_2/1)\cdots(p'_j/1)]\prod_{i=1}^{k}(q_i/1).$$

Hence $S^{-1}A$ is factorial as well. $\square$

We finish the whole post by a comprehensive proposition:

(Proposition 12) Let $A$ be a factorial ring and $p$ a prime element, $\mfk{p}=Ap$. The localization of $A$ at $\mfk{p}$ is principal.

Proof. For $a/s \in S^{-1}A$, we see $p$ does not divide $s$ since if $s = rp$ for some $r \in A$, then $s \in \mfk{p}$, contradicting the fact that $S = A \setminus \mfk{p}$. Since $A$ is factorial, we may write $a = cp^n$ for some $n \geq 0$ and $p$ does not divide $c$ as well (which gives $c \in S$. Hence $a/s = (c/s)(p^n/1)$. Note $(c/s)(s/c)=1/1$ and therefore $c/s$ is a unit. For every $a/s \in S^{-1}A$ we may write it as

$$a/s = u(p^n/1),$$

where $u$ is a unit of $S^{-1}A$.

Let $I$ be any ideal in $S^{-1}A$, and

$$m = \min\{n:u(p^n/1) \in I, u \text{ is a unit }\}.$$

Let’s discuss the relation between $S^{-1}A(p^m/1)$ and $I$. First we see $S^{-1}A(p^m/1)=S^{-1}A(up^m/1)$ since if $v$ is the inverse of $u$, we get

$$vS^{-1}A(up^m/1)=S^{-1}A(p^m/1) \subset S^{-1}A(up^m/1), \\ S^{-1}A(up^m/1)=uS^{-1}A(p^m/1)\subset S^{-1}A(p^m/1).$$

Any element of $S^{-1}A(up^m/1)$ is of the form

$$vup^{m+k}/1=v(p^k/1)up^m/1.$$

Since $up^m/1 \in I$, we see $vup^{m+k}/1 \in I$ as well, hence $S^{-1}A(up^m/1) \subset I$. On the other hand, any element of $I$ is of the form $wup^{m+n}/1=w(p^n/1)u(p^m/1)$ where $w$ is a unit and $n \geq 0$. This shows that $vup^{m+n}/1 \in S^{-1}A(up^m/1)$. Hence $S^{-1}A(p^m/1)=S^{-1}A(up^m/1)=I$ as we wanted. $\square$

Free group

Let $A$ be an abelian group. Let $(e_i)_{i \in I}$ be a family of elements of $A$. We say that this family is a basis for $A$ if the family is not empty, and if every element of $A$ has a unique expression as a linear expression

$$x = \sum_{i \in I} x_i e_i$$

where $x_i \in \mathbb{Z}$ and almost all $x_i$ are equal to $0$. This means that the sum is actually finite. An abelian group is said to be free if it has a basis. Alternatively, we may write $A$ as a direct sum by

$$A \cong \bigoplus_{i \in I}\mathbb{Z}e_i.$$

Free abelian group generated by a set

Let $S$ be a set. Say we want to get a group out of this for some reason, so how? It is not a good idea to endow $S$ with a binary operation beforehead since overall $S$ is merely a set. We shall generate a group out of $S$ in the most freely way.

Let $\mathbb{Z}\langle S \rangle$ be the set of all maps $\varphi:S \to \mathbb{Z}$ such that, for only a finite number of $x \in S$, we have $\varphi(x) \neq 0$. For simplicity, we denote $k \cdot x$ to be some $\varphi_0 \in \mathbb{Z}\langle S \rangle$ such that $\varphi_0(x)=k$ but $\varphi_0(y) = 0$ if $y \neq x$. For any $\varphi$, we claim that $\varphi$ has a unique expression

$$\varphi=k_1 \cdot x_1 + k_2 \cdot x_2 + \cdots + k_n \cdot x_n.$$

One can consider these integers $k_i$ as the order of $x_i$, or simply the time that $x_i$ appears (may be negative). For $\varphi\in\mathbb{Z}\langle S \rangle$, let $I=\{x_1,x_2,\cdots,x_n\}$ be the set of elements of $S$ such that $\varphi(x_i) \neq 0$. If we denote $k_i=\varphi(x_i)$, we can show that $\psi=k_1 \cdot x_1 + k_2 \cdot x_2 + \cdots + k_n \cdot x_n$ is equal to $\varphi$. For $x \in I$, we have $\psi(x)=k$ for some $k=k_i\neq 0$ by definition of the ‘$\cdot$’; if $y \notin I$ however, we then have $\psi(y)=0$. This coincides with $\varphi$. $\blacksquare$

By definition the zero map $\mathcal{O}=0 \cdot x \in \mathbb{Z}\langle S \rangle$ and therefore we may write any $\varphi$ by

$$\varphi=\sum_{x \in S}k_x\cdot x$$

where $k_x \in \mathbb{Z}$ and can be zero. Suppose now we have two expressions, for example

$$\varphi=\sum_{x \in S}k_x \cdot x=\sum_{x \in S}k_x'\cdot x$$

Then

$$\varphi-\varphi=\mathcal{O}=\sum_{x \in S}(k_x-k'_x)\cdot x$$

Suppose $k_y - k_y’ \neq 0$ for some $y \in S$, then

$$\mathcal{O}(y)=k_y-k_y'\neq 0$$

which is a contradiction. Therefore the expression is unique. $\blacksquare$

This $\mathbb{Z}\langle S \rangle$ is what we are looking for. It is an additive group (which can be proved immediately) and, what is more important, every element can be expressed as a ‘sum’ associated with finite number of elements of $S$. We shall write $F_{ab}(S)=\mathbb{Z}\langle S \rangle$, and call it the free abelian group generated by $S$. For elements in $S$, we say they are free generators of $F_{ab}(S)$. If $S$ is a finite set, we say $F_{ab}(S)$ is finitely generated.

An abelian group is free if and only if it is isomorphic to a free abelian group $F_{ab}(S)$ for some set $S$.

Proof. First we shall show that $F_{ab}(S)$ is free. For $x \in M$, we denote $\varphi = 1 \cdot x$ by $[x]$. Then for any $k \in \mathbb{Z}$, we have $k[x]=k \cdot x$ and $k[x]+k’[y] = k\cdot x + k’ \cdot y$. By definition of $F_{ab}(S)$, any element $\varphi \in F_{ab}(S)$ has a unique expression

$$\varphi = k_1 \cdot x_1 + \cdots + k_n \cdot x_n =k_1[x_1]+\cdots+k_n[x_n]$$

Therefore $F_{ab}(S)$ is free since we have found the basis $([x])_{x \in S}$.

Conversely, if $A$ is free, then it is immediate that its basis $(e_i)_{i \in I}$ generates $A$. Our statement is therefore proved. $\blacksquare$

The connection between an arbitrary abelian group an a free abelian group

(Proposition 1) If $A$ is an abelian group, then there is a free group $F$ which has a subgroup $H$ such that $A \cong F/H$.

Proof. Let $S$ be any set containing $A$. Then we get a surjective map $\gamma: S \to A$ and a free group $F_{ab}(S)$. We also get a unique homomorphism $\gamma_\ast:F_{ab}(S) \to A$ by

$$\begin{aligned} \gamma_\ast:F_{ab}(S) &\to A \\ \varphi=\sum_{x \in S}k_x\cdot x &\mapsto \sum_{x \in S}k_x\gamma(x) \end{aligned}$$

which is also surjective. By the first isomorphism theorem, if we set $H=\ker(\gamma_\ast)$ and $F_{ab}(S)=F$, then

$$F/H \cong A.$$

$\blacksquare$

(Proposition 2) If $A$ is finitely generated, then $F$ can also be chosen to be finitely generated.

Proof. Let $S$ be the generator of $A$, and $S’$ is a set containing $S$. Note if $S$ is finite, which means $A$ is finitely generated, then $S’$ can also be finite by inserting one or any finite number more of elements. We have a map from $S$ and $S’$ into $F_{ab}(S)$ and $F_{ab}(S’)$ respectively by $f_S(x)=1 \cdot x$ and $f_{S’}(x’)=1 \cdot x’$. Define $g=f_{S’} \circ \lambda:S’ \to F_{ab}(S)$ we get another homomorphism by

$$\begin{aligned} g_\ast:F_{ab}(S') &\to F_{ab}(S) \\ \varphi'=\sum_{x \in S'}k_{x} \cdot x &\mapsto \sum_{x \in S'}k_{x}\cdot g(x) \end{aligned}$$

This defines a unique homomorphism such that $g_\ast \circ f_{S’} = g$. As one can also verify, this map is also surjective. Therefore by the first isomorphism theorem we have

$$A \cong F_{ab}(S) \cong F_{ab}(S')/\ker(g_\ast)$$

$\blacksquare$

It’s worth mentioning separately that we have implicitly proved two statements with commutative diagrams:

(Proposition 3 | Universal property) If $g:S \to B$ is a mapping of $S$ into some abelian group $B$, then we can define a unique group-homomorphism making the following diagram commutative:

(Proposition 4) If $\lambda:S \to S$ is a mapping of sets, there is a unique homomorphism $\overline{\lambda}$ making the following diagram commutative:

(In the proof of Proposition 2 we exchanged $S$ an $S’$.)

The Grothendieck group

(The Grothendieck group) Let $M$ be a commutative monoid written additively. We shall prove that there exists a commutative group $K(M)$ with a monoid homomorphism

$$\gamma:M \to K(M)$$

satisfying the following universal property: If $f:M \to A$ is a homomorphism from $M$ into a abelian group $A$, then there exists a unique homomorphism $f_\gamma:K(M) \to A$ such that $f=f_\gamma\circ\gamma$. This can be represented by a commutative diagram:

Proof. There is a commutative diagram describes what we are doing.

Let $F_{ab}(M)$ be the free abelian group generated by $M$. For $x \in M$, we denote $1 \cdot x \in F_{ab}(M)$ by $[x]$. Let $B$ be the group generated by all elements of the type

$$[x+y]-[x]-[y]$$

where $x,y \in M$. This can be considered as a subgroup of $F_{ab}(M)$. We let $K(M)=F_{ab}(M)/B$. Let $i=x \to [x]$ and $\pi$ be the canonical map

$$\pi:F_{ab}(M) \to F_{ab}(M)/B.$$

We are done by defining $\gamma: \pi \circ i$. Then we shall verify that $\gamma$ is our desired homomorphism satisfying the universal property. For $x,y \in M$, we have $\gamma(x+y)=\pi([x+y])$ and $\gamma(x)+\gamma(y) = \pi([x])+\pi([y])=\pi([x]+[y])$. However we have

$$[x+y]-[x]-[y] \in B,$$

which implies that

$$\gamma(x)+\gamma(y)=\pi([x]+[y])=\pi([x+y]) = \gamma(x+y).$$

Hence $\gamma$ is a monoid-homomorphism. Finally the universal property. By proposition 3, we have a unique homomorphism $f_\ast$ such that $f_\ast \circ i = f$. Note if $y \in B$, then $f_\ast(y) =0$. Therefore $B \subset \ker{f_\ast}$ Therefore we are done if we define $f_\gamma(x+B)=f_\ast (x)$. $\blacksquare$

Comments of the proof

Why such a $B$? Note in general $[x+y]$ is not necessarily equal to $[x]+[y]$ in $F_{ab}(M)$, but we don’t want it to be so. So instead we create a new equivalence relation, by factoring a subgroup generated by $[x+y]-[x]-[y]$. Therefore in $K(M)$ we see $[x+y]+B = [x]+[y]+B$, which finally makes $\gamma$ a homomorphism. We use the same strategy to generate the tensor product of two modules later. But at that time we have more than one relation to take care of.

Cancellative monoid

If for all $x,y,z \in M$, $x+y=x+z$ implies $y=z$, then we say $M$ is a cancellative monoid, or the cancellation law holds in $M$. Note for the proof above we didn’t use any property of cancellation. However we still have an interesting property for cancellation law.

(Theorem) The cancellation law holds in $M$ if and only if $\gamma$ is injective.

Proof. This proof involves another approach to the Grothendieck group. We consider pairs $(x,y) \in M \times M$ with $x,y \in M$. Define

$$(x,y) \sim (x',y') \iff \exists \ell \in M, y+x'+\ell=x+y'+\ell.$$

Then we get a equivalence relation (try to prove it yourself!). We define the addition component-wise, that is, $(x,y)+(x’,y’)=(x+x’,y+y’)$, then the equivalence classes of pairs form a group $A$, where the zero element is $[(0,0)]$. We have a monoid-homomorphism

$$f:x \mapsto [(x,0)].$$

If cancellation law holds in $M$, then

$$\begin{aligned} f(x) = f(y) &\implies [(x,0)] = [(y,0)] \\ &\implies 0+y+\ell=x+0+\ell \\ &\implies x=y. \end{aligned}$$

Hence $f$ is injective. By the universal property of the Grothendieck group, we get a unique homomorphism $f_\gamma$ such that $f_\gamma \circ \gamma = f$. If $x \neq 0$ in $M$, then $f_\gamma \circ \gamma(x) \neq 0$ since $f$ is injective. This implies $\gamma(x) \neq 0$. Hence $\gamma$ is injective.

Conversely, if $\gamma$ is injective, then $i$ is injective (this can be verified by contradiction). Then we see $f=f_\ast \circ i$ is injective. But $f(x)=f(y)$ if and only if $x+\ell = y+\ell$, hence $x+ \ell = y+ \ell$ implies $x=y$, the cancellation law holds on $M$.

Examples

Our first example is $\mathbb{N}$. Elements of $F_{ab}(\mathbb{N})$ are of the form

$$\varphi=k_1 \cdot n_1 + k_2 \cdot n_2+\cdots + k_m \cdot n_m.$$

For elements in $B$ they are generated by

$$\varphi=1\cdot (m+n)-1\cdot m - 1\cdot n$$

which we wish to represent $0$. Indeed, $K(\mathbb{N}) \simeq \mathbb{Z}$ since if we have a homomorphism

$$\begin{aligned} f:K(\mathbb{N}) &\to \mathbb{Z} \\ \sum_{j=1}^{m}k_j \cdot n_j +B &\mapsto \sum_{j=1}^{m}k_j n_j. \end{aligned}$$

For $r \in \mathbb{Z}$, we see $f(1 \cdot r+B)=r$. On the other hand, if $\sum_{j=1}^{m}k_j \cdot n_j \not\in B$, then its image under $f$ is not $0$.

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In the first example we ‘granted’ the natural numbers ‘subtraction’. Next we grant the division on multiplicative monoid.

Consider $M=\mathbb{Z} \setminus 0$. Now for $F_{ab}(M)$ we write elements in the form

$$\varphi={}^{k_1}n_1{}^{k_2}n_2\cdots{}^{k_m}n_m$$

which denotes that $\varphi(n_j)=k_j$ and has no other differences. Then for elements in $B$ they are generated by

$$\varphi = {}^1(n_1n_2){}^{-1}(n_1)^{-1}(n_2)$$

which we wish to represent $1$. Then we see $K(M) \simeq \mathbb{Q} \setminus 0$ if we take the isomorphism

$$\begin{aligned} f:K(M) &\to \mathbb{Q} \setminus 0 \\ \left(\prod_{j=1}^{m}{}^{k_j}n_j\right)B &\mapsto \prod_{j=1}^{m}n_j^{k_j}. \end{aligned}$$

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Of course this is not the end of the Grothendieck group. But for further example we may need a lot of topology background. For example, we have the topological $K$-theory group of a topological space to be the Grothendieck group of isomorphism classes of topological vector bundles. But I think it is not a good idea to post these examples at this timing.