The ring of real trigonometric polynomials
The ringThroughout we consider the polynomial ring R=\mathbb{R}[\cos{x},\sin{x}].This ring has a lot of non-trivial properties which give us a good chance to study commutative ring theory.
Characters in Analysis and Algebra
In this post, we study the concept of character, what it is about in abstract harmonic analysis and how to use it Galois theory.
Dedekind Domain and Properties in an Elementary Approach
You can find contents about Dedekind domain (or Dedekind ring) in almost all algebraic number theory books. But many properti...
Tensor Product as a Universal Object (Category Theory & Module Theory)
IntroductionIt is quite often to see direct sum or direct product of groups, modules, vector spaces. Indeed, for modules over...
Why Does a Vector Space Have a Basis (Module Theory)
Module and vector spaceFirst we recall some backgrounds. Suppose $A$ is a ring with multiplicative identity $1_A$. A left mod...
Rings of Fractions and Localisation
Is perhaps the most important technical tools in commutative algebra. In this post we are covering definitions and simple pro...
The Grothendienck Group
Free groupLet $A$ be an abelian group. Let $(e_i)_{i \in I}$ be a family of elements of $A$. We say that this family is a bas...
A long exact sequence of cohomology groups (zig-zag and diagram-chasing)
Exterior differentiation(This section is intended to introduce the background. Feel free to skip if you already know exterior...
Cauchy sequence in group theory
Recall - Cauchy sequence in analysisBefore we go into group theory, let’s recall how Cauchy sequence is defined in analysis. ...