Ideals: Let be any ring and be a subring of , then is said to be a right ideal of if and a left ideal of if .
Thus a non-empty subset of is said to be an ideal of if:
(i) is a subgroup of under addition.
(ii) For all and , both and .
Proper Ideals: If is a commutative ring with unity and , the ideal is called the principal ideal generated by and is denoted by . Thus stands for the ideal generated by .
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 be a commutative ring. An ideal of ring is said to be a prime ideal of if
Example: In the commutative ring of integers , the ideal is a prime ideal since if , then and consequently or as is prime.