Definition(Ring)
A non-empty set R, equipped with two binary operations, called addition (+) and multiplication (.) is called a ring if the following postulates are satisfied.
(1) R is an additive Abelian group,
(2) R is an multiplicative semigroup
(3) The two distributive laws hold good, viz
a. (b + c) =a.b + a.c. (left distributive law)
(a + b). c = a.c + b.c, (right distributive law) for all a, b, c R
Note:
1. In order to be a ring, there must be a non-empty set equipped with two binary operations, viz, + and . , satisfying the postulates mentioned above. Later on we will simply write ‘‘Let R be a ring’’. It should be borne in mind that there are two binary operations, viz, + and , on R.
2. It should also be borne in mind that the binary operations + and . may not be own usual addition and multiplication.
3. Since R is an additive Abelian group, the additive identity, denoted by 0, belongs to R. It is called the zero element of R. Here 0 is a symbol.
It should not be confused with the real number zero.
4. If a , bR, then – bR ( R is an additive group). a+ (– b) is, generally, denoted by a – b.
5. a.b is, generally, written as ab.
Commutative Ring: R is commutative if ab = ba, for all a, bR.
Ring with unity : R is a ring with unity (or identity) if there exists an element e R such that
ae = a = ea, for all a R.
Note: A ring may or may not have a unity.
2. If R is a ring with unity, unity may be same as the zero element. For example, {0} is a commutative R with unity. Here 0 is the additive as well as the multiplicative identity. It is the zero ring.
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