# Self-injective ring

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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.

#### References

 [1] L.A. Skornyaka, A.V. Mikhalev, "Modules" Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 14 (1976) pp. 57–190 (In Russian) [2] C. Faith, "Algebra" , 1–2 , Springer (1973–1976) [3] J. Lawrence, "A countable self-injective ring is quasi-Frobenius" Proc. Amer. Math. Soc. , 65 : 2 (1977) pp. 217–220

An essential right ideal of a ring $R$ is an ideal $E$ such that $E \cap I \neq 0$ for all non-zero right ideals $I$ of $R$. In a right Ore domain (cf. below) every non-zero right ideal is essential. Let ${\mathcal E} ( R)$ be the set of essential right ideals of $R$;

$$\zeta ( R) = \{ {a \in R } : {a E = 0 \textrm{ for some } E \in {\mathcal E} ( R) } \}$$

is an ideal, called the right singular ideal of $R$.

Let $S$ be the multiplicatively closed subset of regular elements of $R$( i.e. non-zero-divisors of $R$). If $S$ satisfies the right Ore condition (cf. Associative rings and algebras), $R$ is called a right Ore ring. A right Ore domain is an integral domain that is a right Ore ring.

#### References

 [a1] J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987) pp. Part I, Chapt. 2
How to Cite This Entry:
Self-injective ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Self-injective_ring&oldid=48650
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article