Nakayama’s lemma

The Nakayama lemma as a concept is present throughout Commutative Algebra. And truth be told, learning it is not easy. The proof contains a small trick that is deceptively simple, but throws off many people. Also, it is easy to dismiss this lemma as unimportant. But as one would surely find out later, this wouldContinue reading “Nakayama’s lemma”

Atiyah-Macdonald Part II

Let me start by first talking about the proof of the fact that the intersection of all prime ideals in a ring is the nilradical. The proof is constructive, and hence perhaps non-trivial. I will attempt to generalize the methodology of the proof. This proof deals with creating a sort of border around the numberContinue reading “Atiyah-Macdonald Part II”

Solutions to some problems from Chapter 1 in Atiyah-Macdonald. And more importantly, insights and generalizations

If , where is the Jacobian ideal of a ring , then is a unit for all . looks like a particularly arbitrary expression. Let us break it down. for all is an alternate representation for . All maximal ideals contain . Now let us suppose that is not a unit. Then is contained inContinue reading “Solutions to some problems from Chapter 1 in Atiyah-Macdonald. And more importantly, insights and generalizations”

Localization of a ring

So what exactly is the localization of a ring? It is creating a field-imitation (and NOT necessarily a field with multiplicative inverses) for every ring. It includes the creation of multiplicative inverse- imitations (for elements that are not zero-divisors). How does it do this? It apes the steps taken to create a field of fractionsContinue reading “Localization of a ring”