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 $ \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

Comments

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