Ideal - Knowlpedia

Let $R$ be a ring and $I\le (R,+)$ an additive subgroup. Then $I$ is a left ideal if $rI\subseteq I$ for every $r\in R$ , a right ideal if $Ir\subseteq I$ for every $r\in R$ , and a two-sided ideal if both conditions hold.

Remarks

Two-sided ideals are exactly the kernels of ring homomorphisms and are the ideals for which the quotient ring $R/I$ is defined. In commutative rings the three notions coincide.

Examples

In $\mathbb Z$ , every ideal has the form $n\mathbb Z$ for some $n\ge 0$ .
In $k[x,y]$ , the set $(x,y)$ of polynomials with zero constant term is an ideal.
In $M_2(k)$ , the set of matrices whose second column is zero is a left ideal but not a right ideal.