2.3 Rings

2.3 Rings

Definition 2.3.1 A ring (R, +, x) consists of a set R with two binary operations arbi-

trarily denoted + (addition) and x (multiplication) on R, satisfying the following axioms.

1. (R, +) is an abelian group with identity denoted 0.

2. The operation x is associative. That is a x (b x c)=(a x b) x c for all a, b, c Є R.

3. There is a multiplicative identity denoted 1, with 1 ≠ 0, such that 1 x a = a x 1=a

for all a Є R.

4. The operation x is distributive over +. That is, a x (b + c)=(a x b)+(a x c) and

(b + c) x a =(b x a)+(c x a) for all a, b, c Є R.

The ring is a commutative ring if a x b = b x a for a, b Є R.

Definition 2.3.2 An element of a of a ring R is called a unit or an invertible element if

there is an element b Є R such that a x b =1.