Let $R$ be a ring with identity. A proper left (right, two-sided) ideal $\mathfrak{m}\subsetneq R$ is said to be maximal if $\mathfrak{m}$ is not a proper subset of any other proper left (right, two-sided) ideal of $R$.

One can prove:

• A left ideal $\mathfrak{m}$ is maximal if and only if $R/\mathfrak{m}$ is a simple left $R$-module.

• A right ideal $\mathfrak{m}$ is maximal if and only if $R/\mathfrak{m}$ is a simple right $R$-module.

• A two-sided ideal $\mathfrak{m}$ is maximal if and only if $R/\mathfrak{m}$ is a simple ring.

All maximal ideals are prime ideals. If $R$ is commutative, an ideal $\mathfrak{m}\subset R$ is maximal if and only if the quotient ring $R/\mathfrak{m}$ is a field.