# 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 Zn 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 Zn 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.