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”

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”

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”