Two-sided ideal

A two-sided ideal of a ring $R$ is an ideal $I\subseteq R$ such that $rI\subseteq I$ and $Ir\subseteq I$ for all $r\in R$.

Two-sided ideals are precisely the ideals for which the quotient ring $R/I$ carries a well-defined multiplication induced from $R$. In commutative rings, every ideal is automatically two-sided.

Examples:

• In $\mathbb Z$, every ideal $n\mathbb Z$ is two-sided.
• In $M_n(k)$, the only two-sided ideals are $\{0\}$ and $M_n(k)$.
• In an upper triangular matrix ring, strictly upper triangular matrices form a two-sided ideal.