We will discuss short exact sequences here. They are of the form , where im = ker . Example: , such that and . What is the point of such a construction? Is there any way to visualize this? Whatever the image of , you want to completely obliterate it. Leave no trace of it.Continue reading “A short note about short exact sequences.”
Monthly Archives: June 2014
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”
I’ve always found the construction of the quotient field of a domain very arduous and time-consuming. Most people get lost somewhere in the proof. I am going to try and make it more transparent. What are we trying to do? We’re trying to convert a ring into a field (please forgive my language). How areContinue reading
Developing new axioms for Elementary Geometry
I have always been bad at geometry. Always. How does one develop geometric intuition? Let us talk about the fundamental building blocks of Geometry- Congruency. I must confess I never really understood HOW the whole of geometry is motivated by a seemingly useless and, let’s face it, mysterious concept of Congruency. Two figures which areContinue reading “Developing new axioms for Elementary Geometry”
VSRP 1: On V(a)
So what exactly are prime ideals? They are ideals such that or . Let , where is a commutative ring. How is the Zariski topology motivated? Let be the set of all prime ideals which contain . All ideals that contain contain the whole of . Hence all members of also contain the whole ofContinue reading “VSRP 1: On V(a)”
I have been been part of the VSRP program for about 10 days. I have solved some problems from Atiyah-Macdonald and some from a couple of other books. I have also brushed a little of topology and other parts of Algebra. In spite of having solved problems on prime ideals and the Zariski Topology, thereContinue reading
TIFR VSRP
I have been at TIFR for a week. Some thoughts: The emphasis is clearly on “what book have you read? I followed this book. Have you attempted the exercises from this book?” This is similar to what I saw at ISI, Kolkata. I did not focus on problem solving while learning Mathematics on my own.Continue reading “TIFR VSRP”
