Ring:-
An algebraic system (R,+,.) consisting of a
non-empty set R with two binary composition (to be denoted by addition and
multiplication) is called a ring if following axiom are satisfied-
(R,+) is
an abelian group.
1- Closure axiom
2- Associative law
3- Identity element
4- Inverse axiom
(R,.) is
a semi group.
1- Closure axiom
2- Associative law
3- Multiplication distributive over
addition i.e.
a.(b+c)=a.b+a.c, ∀ a, b, c∊R
and (b+c).a=b.a+c.a, ∀a, b, c∊R
Commutative Ring:-
If
multiplication is commutative in R then it is called commutative ring, i.e.
a.b=b.a ,∀ a,b∊R
Ring with unity element:-
If multiplicative identity ( called unity
element of R and denoted by 1) exist in R then it is called ring with unity
i.e. 1.a=a.1=a ,∀a∊R
Ring without zero divisors:-
If a,b are any two element of R such that aob=0 ⇨ a=0 or
b=0,
then R is said to be without zero divisors.
Ring with zero devisors:-
If the
product of two non-zero elements a, b of R is zero then it is called ring with
zero divisors. i.e.
a≠0, b≠0 and ab=0
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