# Ideals in Rings

__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.