# Self-injective ring

*left*

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 are self-injective rings. If is a self-injective ring with Jacobson radical , then the quotient ring 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 -modules are self-injective rings if and only if is quasi-Frobenius. If is the cogenerator of the category of left -modules, then is a self-injective ring. If the singular ideal of a ring is zero, then its injective hull can be made into a self-injective ring in a natural way. A group ring is left self-injective if and only if is a self-injective ring and is a finite group. The direct product of self-injective rings is self-injective. A ring is isomorphic to the direct product of complete rings of linear transformations over fields if and only if 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 |

#### Comments

An essential right ideal of a ring is an ideal such that for all non-zero right ideals of . In a right Ore domain (cf. below) every non-zero right ideal is essential. Let be the set of essential right ideals of ;

is an ideal, called the right singular ideal of .

Let be the multiplicatively closed subset of regular elements of (i.e. non-zero-divisors of ). If satisfies the right Ore condition (cf. Associative rings and algebras), 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 |

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Self-injective ring.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Self-injective_ring&oldid=17113