Chinese Remainder Theorem in Several Scenarios of Ring Theory
We study the Chinese remainder theorem in various contexts and abstract levels.Every Regular Local Ring is Cohen-Macaulay
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.Hensel's Lemma - A Fair Application of Newton's Method and 'Double Induction'
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 $\mathbb{Q}_p$.The ring of trigonometric polynomials with complex scalars
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 ring of real trigonometric polynomials
The ring
Throughout we consider the polynomial ring
This ring has a lot of non-trivial properties which give us a good chance to study commutative ring theory.