Vol. 40, No. 3, 1972

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Minimal injective cogenerators for the class of modules of zero singular submodule

Vasily Cateforis

Vol. 40 (1972), No. 3, 527–539

Let R be a ring (with 1) of zero singular right ideal and let Q be its maximal right quotient ring; let 𝒩 be the class or all (unitary) right R-modules of zero singular submodule. An element M of 𝒩 is said to be an injective cogenerator for 𝒩 if M is an injective module and every element of 𝒩 can be embedded in a direct product of copies of M;M is said to be a minimal injective cogenerator for 𝒩 if M is the only direct summand of M, which is an injective cogenerator for 𝒩. This paper deals with the question of existence and uniqueness of a minimal injective cogenerator for 𝒩 (and in 𝒩). If a minimal injective cogenerator for 𝒩 exists, then it is (a) isomorphic to a minimal faithful direct summand of Q, (b) isomorphic to a direct summand of every injective cogenerator for 𝒩 (and in 𝒩) and (c) unique (up to isomorphism). Whether Q is (or is not) a prime ring, affects the structure, though not the existence, of a minimal injective cogenerator for 𝒩: a minimal injective cogenerator for 𝒩, if it exists, is (up to isomorphism) a faithful minimal right ideal of Q iff Q is a prime ring and so in this case Q is a minimal injective cogenerator 𝒩 iff Q is a division ring. On the other hand, if RR is finite dimensional (Goldie) then a minimal injective cogenerator for 𝒩 exists; it is Q iff Q is (ring) isomorphic to a finite product of division rings.

Mathematical Subject Classification
Primary: 16A52
Received: 15 September 1971
Revised: 11 November 1971
Published: 1 March 1972
Vasily Cateforis