We study the Chinese remainder theorem in various contexts and abstract levels.

In this post we show that the class of regular local rings (the abstract version of power series rings) is a subclass of Cohen-Macaulay ring.

We prove the celebrated Hensel's lemma using the so-called Newton's method and "double induction", and try to find solutions of polynomials in $\mathbb{Q}_p$.

This post is a continuation of a previous post about the ring of trigonometric polynomials over the real field. Now we have jumped into the complex field, and the extension is not a trivial matter.

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.

You can find contents about Dedekind domain (or Dedekind ring) in almost all algebraic number theory books. But many properti...

IntroductionIt is quite often to see direct sum or direct product of groups, modules, vector spaces. Indeed, for modules over...

Is perhaps the most important technical tools in commutative algebra. In this post we are covering definitions and simple pro...