Today we shall discuss some interesting properties of elements of a ring. 1. If is not a unit, then it is present in some maximal ideal of the ring . Self-explanatory. 2. If is present in every maximal ideal, then is a unit for all . Proof: Let not be a unit. Then it isContinue reading “Some facts, better explained, from Atiyah-Macdonald”

# Monthly Archives: October 2013

## The existence or inexistence of a maximal element

Have you ever wondered why the real number line does not have a maximal element?Take . Define an element . Declare that is greater than any element in in . Can we do that? Surely! We’re defining it thus. In fact, does not even have to be a real number! It can just be someContinue reading “The existence or inexistence of a maximal element”

## Breaking down Zorn’s lemma

Today I’m going to talk about Zorn’s lemma. No. I’m not going to prove that it is equivalent ot the Axiom of Choice. All I’m going to do is talk about what it really is. Hopefully, I shal be able to create a visually rich picture so that you may be able to understand itContinue reading “Breaking down Zorn’s lemma”

## The mysterious linear bounded mapping

What exactly is a linear bounded mapping? The definition says is called a linear bounded mapping if . When you hear the word “bounded”, the first thing that strikes you is that the mappings can’t exceed a particular value. That all image points are within a finite outer covering. That, unfortunately, is not implied byContinue reading “The mysterious linear bounded mapping”

## The “supremum” norm

Today I shall speak on a topic that I feel is important. All of us have encountered the “” norm in Functional Analysis. In , for . In the dual space , . What is the utility of this “sup” norm? Why can’t our norm be based on “inf” or the infimum? Or even ?Continue reading “The “supremum” norm”

## On making a choice between hypotheses

At 15:33, Peter Millican says “How can any criterion of reliable knowledge be chosen, unless we already have some reliable criterion for making that choice?” What does this actually mean? Say I have two hypotheses- A and B. One of them is true, whilst the other is false. But I don’t know which is which.Continue reading “On making a choice between hypotheses”

## The factoring of polynomials

This article is about the factorization of polynomials: First, I’d like to discuss the most important trick that is used directly or implicitly in most of the theorems given below. Let us consider the polynomial . Let be the first coefficient, starting from , not to be a multiple of in . Let be theContinue reading “The factoring of polynomials”

## Proving J[i] is a Euclidean ring.

Today we’ll try to realign our intuition with the standard textbook proof of “ is a Euclidean ring”. denotes the set of all complex numbers of the form where and are integers. . Let us take two complex numbers . Let and , where and . Then if there exists a complex number such thatContinue reading “Proving J[i] is a Euclidean ring.”

## Euclidean rings and prime factorization

Now we will talk about the factorization of elements in Euclidean rings. On pg.146 of “Topics in Algebra” by Herstein, it says: “Let be a Euclidean ring. Then every element in is either a unit in or can be written as the product of a finite number of prime elements in .” This seems elementary.Continue reading “Euclidean rings and prime factorization”

## Euclidean rings and generators of ideals

This is to address a point that has just been glazed over in “Topics in Algebra” by Herstein. In a Euclidean ring, for any two elements such that . Also, there exists a function such that . We also know that the element with the lowest d-value generates the whole ring . The proof ofContinue reading “Euclidean rings and generators of ideals”