Editing Injective module (section)

===Self-injective rings===
Every ring with unity is a [[free module]] and hence is a [[projective module|projective]] as a module over itself, but it is rarer for a ring to be injective as a module over itself, {{harv|Lam|1999|loc=§3B}}. If a ring is injective over itself as a right module, then it is called a right self-injective ring. Every [[Frobenius algebra]] is self-injective, but no [[integral domain]] that is not a [[field (mathematics)|field]] is self-injective. Every proper [[quotient ring|quotient]] of a [[Dedekind domain]] is self-injective.

A right [[Noetherian ring|Noetherian]], right self-injective ring is called a [[quasi-Frobenius ring]], and is two-sided [[Artinian ring|Artinian]] and two-sided injective, {{harv|Lam|1999|loc=Th. 15.1}}.  An important module theoretic property of quasi-Frobenius rings is that the projective modules are exactly the injective modules.