Vol. 40, No. 3, 1972

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 332: 1
Vol. 331: 1  2
Vol. 330: 1  2
Vol. 329: 1  2
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Online Archive
Volume:
Issue:
     
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Officers
 
Subscriptions
 
ISSN 1945-5844 (electronic)
ISSN 0030-8730 (print)
 
Special Issues
Author index
To appear
 
Other MSP journals
Minimal injective cogenerators for the class of modules of zero singular submodule

Vasily Cateforis

Vol. 40 (1972), No. 3, 527–539
Abstract

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
Milestones
Received: 15 September 1971
Revised: 11 November 1971
Published: 1 March 1972
Authors
Vasily Cateforis