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

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

Something that has confused me for long is the condition for injectivity for a homomorphism. The condition is that the kernel should just be the identity element. I used to think that maybe this condition for injectivity applies to all mappings, and wondered why I hadn’t come across this earlier. No. This condition applies onlyContinue reading

## Integral domains and characteristics

Today we shall talk about the characteristic of an integral domain, concentrating mainly on misconceptions and important points. An integral domain is a commutative ring with the property that if and , then . Hence, if , then or (or both). The characteristic of an integral domain is the lowest positive integer such that .Continue reading “Integral domains and characteristics”

Today we will discuss compactness in the metric setting. Why metric? Because metric spaces lend themselves more easily to visualisation than other spaces. Let us imagine a metric space with points scattered all over it. If we can find an infinite number of such points and construct disjoint open sets centred on them, then cannotContinue reading

Today I plan to write a treatise on spaces. are normed spaces over with the p-norm, or . Say we have the space over . This just means that , where . That is a norm is proved using standard arguments (including Minkowski’s argument, which is non-trivial). Now we have a metric in  spaces: .Continue reading

Let be a mapping. We will prove that , with equality when is injective. Note that does not have to be closed, open, or even continuous for this to be true. It can be any mapping. Let . The mapping of in is . As for , it may overlap with , we the mappingContinue reading

Today, I will discuss this research paper by Javed Ali, Professor of Topology and Analysis, BITS Pilani. What exactly is a proximinal set? It is the set of elements in which for any , you can find the nearest point(s) to it in . More formally, for each such that . This article says aContinue reading

This is the second time I’m checking whether I can write a LATEX equation in wordpress.