# Rings

- 2.175 Definition A
*ring (R,+,x)*consists of a set*R*with two binary operations arbitrarily denoted + (addition) and x (multiplication) on /?, satisfying the following axioms. - (i) (Я, +) is an abelian group with identity denoted 0.

- (ii) The operation x is associative. That is,
*а*x*(b*x с) = (a x*b)*x*c*for all*a,b,c*6*R.* - (iii) There is a multiplicative identity denoted 1, with 1^0, such that lxa. = axl = o for all
*a**e**R.* - (iv) The operation x is
*distributive*over +. That is,*a*x*(b*+*c)*=*(a*x*b) + (a*x c) and (6 + c) x*a = (b*x*a)*+ (c x*a)*for all*a,b,c*6*R.*

The ring is a *commutative ring* if *a x b = b x a* for all *a,b € R.*

2.176 Example The set of integers Z with the usual operations of addition and multiplication is

a commutative ring. □

- 2.177 Example The set Z„ with addition and multiplication performed modulo
*n*is a commutative ring. □ - 2.178 Definition An element
*a*of a ring*R*is called a*unit*or an*invertible element*if there is an element*b**e**R*such that*a*x*b =*1. - 2.179 Fact The set of units in a ring
*R*forms a group under multiplication, called the*group of units*of*R.* - 2.180 Example The group of units of the ring Z
_{n}is Z* (see Definition 2.124). □

# Fields

2.181 Definition A *field* is a commutative ring in which all non-zero elements have multiplicative inverses.

m times

/-^-X

- 2.182 Definition The
*characteristic*of a field is 0 if 1 + 1 + • ■ • + 1 is never equal to 0 for any m > 1. Otherwise, the characteristic of the field is the least positive integer*m*such that £]'=i 1 equals 0. - 2.183 Example The set of integers under the usual operations of addition and multiplication is

not a field, since the only non-zero integers with multiplicative inverses are 1 and—1. However, the rational numbers Q, the real numbers R, and the complex numbers C form fields of characteristic 0 under the usual operations. □

- 2.184 Fact Z„ is a field (under the usual operations of addition and multiplication modulo n) if and only if
*n*is a prime number. If*n*is prime, then Z_{n}has characteristic n. - 2.185 Fact If the characteristic
*m*of a field is not 0, then*m*is a prune number. - 2.186 Definition A subset
*F*of a field*E*is a*subfield*of*E*if*F*is itself a field with respect to the operations of*E.*If this is the case,*E*is said to be an*extension field*of*F.*