# maximal ideal

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.

 Title maximal ideal Canonical name MaximalIdeal Date of creation 2013-03-22 11:50:57 Last modified on 2013-03-22 11:50:57 Owner djao (24) Last modified by djao (24) Numerical id 8 Author djao (24) Entry type Definition Classification msc 13A15 Classification msc 16D25 Classification msc 81R50 Classification msc 46M20 Classification msc 18B40 Classification msc 22A22 Classification msc 46L05 Related topic ProperIdeal Related topic Module Related topic Comaximal Related topic PrimeIdeal Related topic EveryRingHasAMaximalIdeal