Some facts, better explained, from Atiyah-Macdonald

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”

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”

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”

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”