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Pure Math

Ring Theory

Introduction

Ring theory is a branch of mathematics that is closely interlinked with group theory. To create a ring, we take a group with its “addition” operation, and add “multiplication” to the mix! We say that this group is a ring if we can apply the distribution law like standard addition and multiplication. If the nonzero elements of the ring form a group under multiplication, then we call the ring a field. It is important to include this “nonzero” part, because 0, or the identity of the group, does not have an inverse under multiplication. Instead, the identity of the multiplicative group is another element, commonly referred to as 1.

Examples

The integers, rationals, and reals are all examples of rings, using standard addition and multiplication. Another example is the set of n by n matrices under matrix addition and multiplication. Note that the rationals and reals are also fields, as we can take the multiplicative inverse of every nonzero element.

Extensions

Rings can contain other rings, including the trivial ring (only containing the additive and multiplicative identity) and the entire ring. These subsets are called subrings if they are rings and subfields if they are fields. 

Exercise: Verify that the subrings of \Z, the integers, are the integer multiples of \Z.

One way to construct rings is to take a ring P and a subring Q, and then construct

    \[P[X] = \{\sum_{k=0}^n f_kX^k\mid f_0,\dots,f_n\in F\}\]

for some element X\in Q. The resulting set is a ring extension of P. If X is algebraic in Q, then P(X) is actually a field extension. Naturally, we could replace X with a set as well, taking P(X) as the set of all linear combinations of elements in X

Exercise: The complex numbers \mathbb C can be expressed as a field extension of the real numbers \R. What is the X in this case?

Applications

Fields and their extensions are widely used in many fields relating to algebra and number theory, including abstract algebra and Galois theory. They are one of the most fundamental mathematical objects, and are a way to describe the overall structure of sets that F_p (integers modulo p), the rationals, and others share.

Prerequisites/What you need to know

Rings do not take a lot of background in mathematics to understand. However, ring theory can be thought of as an expansion to group theory that adds on an additional binary operation to the set.

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Comprehensive look at ring theory

Concise overview of theorems and definitions