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

Example: In 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.