Ideals in Rings

Ideals: Let \left( {R, + , \times } \right) be any ring and S be a subring of R, then S is said to be a right ideal of R if a \in S,\,b \in S\, \Rightarrow ab \in S and a left ideal of R if a \in S,\,b \in S\, \Rightarrow ba \in S.

Thus a non-empty subset S of R is said to be an ideal of R if:

(i) S is a subgroup of R under addition.

(ii) For all a \in S and b \in R, both ab and ba \in S.

Proper Ideals: If R is a commutative ring with unity and a \in R, the ideal \left\{ {ax:x \in R} \right\} is called the principal ideal generated by a and is denoted by \left( a \right).  Thus \left( a \right) stands for the ideal generated by a.

Principal Ideal Ring: A commutative ring with unity for which every ideal is a principal ideal is said to be a principal ideal ring.

Prime Ideal: Let R be a commutative ring. An ideal P of ring R is said to be a prime ideal of R if

ab \in P,\,\,\,a,b \in R \Rightarrow a \in P\,\,\,{\text{or}}\,\,\,b \in P

ExampleIn the commutative ring of integers I, the ideal P = \left\{ {5r:r \in I} \right\} is a prime ideal since if ab \in P, then 5|ab and consequently 5|a or 5|b as 5 is prime.