Ideal

An ideal in a ring $R$ is an additive subgroup $I\le (R,+)$ such that it is stable under multiplication by elements of $R$ on one side:

• $I$ is a left ideal if $rI\subseteq I$ for all $r\in R$,
• $I$ is a right ideal if $Ir\subseteq I$ for all $r\in R$.

Ideals are exactly the kernels of ring homomorphisms, and they are the congruence data needed to form a quotient ring . In commutative rings the left/right distinction disappears, but in noncommutative rings it is essential.

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.