# 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.