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subring,ideal

 Def:

 Subring : 

Let R be a ring. A non-empty subset S of R is said to be a subring of R if S itself is a ring under the same binary operations as in R.

Any subring S of R is said to be a proper subring of R if S ¹R . Also, {0} is known as trivial subring.

Result :

A non-empty subset S of a ring R is a subring of R if and only if (i) a - b  Î S for all a, b Î S i.e. S is an additive subgroup of R. (ii) ab  Î S for all a, b Î S.

 Left ideal :

 A non – empty subset S of a ring R is called a left ideal of R if

(i)                  a- b Î S for all a, b  ÎS

(ii)                (ii) ra  ÎS for all a  Î S , r  ÎR .

 Right ideal : 

A non – empty subset S of a ring R is called a right ideal of R if 

(i) a- b  ÎS for all a, b  Î S

(ii)               ar  ÎS for all a  ÎS , r  ÎR .

Remarks :

1.       In a commutative ring every left ideal or right ideal is a both-sided ideal.

2.       An ideal A of a ring R is called a proper ideal if A  ¹R . Further  , zero ideal {0} is also called trivial ideal or zero ideal.  

3.       Every left ideal, right ideal and ideal is always a subring.

4.       Every subring need not be an ideal.

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