جبر الحلقات- المحاضرة15

Definition 1.1 Given an algebraic system (S, ? , ?), the operation ? is said to be left distributive over ? if a ? (b?c) = (a?b) ? (a?c)
and right distributive if (b?c) ? a = (b?a) ? (c ?a)
for all elements a, b, c S. The operation ? is distributive over ? if both of these conditions hold.
Definition 1.2 A ring is an ordered triple ( R, +, ? ) consisting of a nonempty set R and two binary operations + and ? defined on R such that

1. (R, +) is a commutative group,
2. (R, ?) is a semigroup, and
3. the operation ? is distributive over the operation + .
A ring ( R, +, ? ) consists of a nonempty set R together with two binary operations + and ? of addition and multiplication on R for which the following conditions are satisfied :
1. a + b = b+ a,
2. (a +b) + c = a +(b +c),
3. there exists an element 0 in R such that a +0=a for every a R,
4. for each a R there exists an element –a R such that a +(-a) =0,
5. (a? b) ?c = a ? (b? c), and
6. a? ( b+ c) = a? b + a? c and (b +c) ? a = b? a + c ? a.
Definition 1.3
1. A commutative ring ( R, +, ? ) in which multiplication is a commutative operation, a? b = b? a for all a, b R. (In case a b= b a for a particular pair a, b we express this fact by saying a and b commute)
2. A ring with identity is a ring ( R, +, ? ) in which there exists an identity element for the operation of multiplication normally represented by the symbol 1, so that a?1= 1? a= a for all a R.
Example 1.1
If Z, Q, R denote the sets of integers, rational, and real numbers, respectively, then (Z, +, ?), (Q, +, ?), (R , +, ?) are examples of rings; here, + and ? are taken to be ordinary addition and multiplication. In each of these cases the ring is commutative and has the integer 1 for an identity element.

Example 1.2
Let X be a nonempty set and P(X) denote the collection of all subsets of X. Both the systems (P(X), ) and (P(X), ) fail to be rings since neither (P(X), ) nor (P(X), ) forms a group. However, (P(X), ?) is a commutative group, where ? indicates the symmetric difference operation A ? B = (A-B) (B-A). Since (P(X), ) is clearly a commutative semigroup and it can be proved that the operation is left distributive over ?. The system (P(X), ?, ) is a commutative ring with identity.

Example 1.3
(Z ) is a commutative ring with identity , where
= { [0], [1], …, [n-1]} is the set of integers modulo n


Example 1.4
Consider the set M of all 2x2 matrices with entries from R . With the usual rules of matrix addition and multiplication, (M ,+, ?) becomes a noncomutative ring with identity.