This chapter introduces the notion of a ring, more specifically, a commutative ring with unity. The theory of rings provides a useful conceptual framework for reasoning about a wide class of interesting algebraic structures. Intuitively speaking, a ring is an algebraic structure with addition and multiplication operations that behave like we expect addition and multiplication should. While there is a lot of terminology associated with rings, the basic ideas are fairly simple.

Definitions, basic properties, and examples

Definition 9.1. A commutative ring with unityis a set R together with addition and multiplication operations on R, such that:

1. (i) the set R under addition forms an abelian group, and we denote the additive identity by 0R;

2. (ii) multiplication is associative; that is, for all a, b, c ∈ R, we have a(bc) = (ab)c;

3. (iii) multiplication distributes over addition; that is, for all a, b, c ∈ R, we have a(b + c) = ab + ac and (b + c)a = ba + ca;

4. (iv) there exists a multiplicative identity; that is, there exists an element 1R ∈ R, such that 1R · a = a = a · 1R for all a ∈ R;

5. (v) multiplication is commutative; that is, for all a, b ∈ R, we have ab = ba.

There are other, more general (and less convenient) types of rings–one can drop properties (iv) and (v), and still have what is called a ring.