High Quality Content by WIKIPEDIA articles! In ring theory, a left primitive ring is a ring which has a faithful simple left module. Examples include matrix algebras over division rings, non-commutative polynomial rings and simple Artinian rings. A ring R is said to be a left primitive ring if and only if it has a faithful simple left R-module. A right primitive ring is defined similarly with right R-modules. By the Jacobson density theorem, a ring is left primitive if and only if it is isomorphic to a dense ring of endomorphisms of a right vector space over a division ring. A commutative ring is left primitive if and only if it is a field. A left artinian ring is left primitive if and only if it is simple if and only if it is prime. A ring is left primitive if and only if it is prime and has a faithful left module of finite length.

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