# Self-injective ring

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
A ring that, as a left module over itself, is injective (cf. Injective module). A right self-injective ring is defined in a symmetric way. The classical semi-simple rings and all rings of residues of integers $\mathbf Z /( n)$ are self-injective rings. If $R$ is a self-injective ring with Jacobson radical $J$, then the quotient ring $R/J$ is a regular ring (in the sense of von Neumann). A regular self-injective ring is continuous. Every countable self-injective ring is quasi-Frobenius (cf. Quasi-Frobenius ring). A left self-injective ring is not necessarily right self-injective. The ring of matrices over a self-injective ring and the complete ring of linear transformations of a vector space over a field are self-injective. The rings of endomorphisms of all free left $R$- modules are self-injective rings if and only if $R$ is quasi-Frobenius. If $M$ is the cogenerator of the category of left $R$- modules, then $\mathop{\rm End} _ {R} M$ is a self-injective ring. If the singular ideal of a ring $R$ is zero, then its injective hull can be made into a self-injective ring in a natural way. A group ring $RG$ is left self-injective if and only if $R$ is a self-injective ring and $G$ is a finite group. The direct product of self-injective rings is self-injective. A ring $R$ is isomorphic to the direct product of complete rings of linear transformations over fields if and only if $R$ is a left self-injective ring without nilpotent ideals for which every non-zero left ideal contains a minimal left ideal.