Rings may well be the most familiar algebraic structure. We all grew up with integers, polynomials, rational and real numbers. These familiar rings do not, however, prepare us for the huge variety of rings and the complexity of ring theory. Rings and their modules should be studied together, and that is what we do in this chapter.

DEFINITIONS

We start fromthe definitions of the objects, the appropriate homomorphisms, and the relevant sub-objects. Since both rings and modules will be in play, we need to do this for both structures.

RINGS

Definition 5.1.1A ring is a set R together with two operations written as addition and multiplication, such that

1. 1. R with addition is an abelian group with neutral element 0

2. 2. R with multiplication is a monoid, i.e., multiplication is associative and there is a neutral element 1

3. 3. multiplication distributes over addition:

a(b + c) = ab + ac and (b + c)a = ba + ca

for all a, b, c ∈ R.

It is useful to have names for elements in a ring with special properties.

Definition 5.1.2Let R be a ring.

1. 1. We say an element x ∈ R is a unit if there exists an element x′ ∈ R such that xx′ = x′x = 1.

2. 2. We say an element x ∈ R is a zero-divisor if x ≠ 0 and there exists y ∈ R, y ≠ 0, such that xy = 0 or yx = 0.

3. […]