International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 3, Pages 507-512

Epis and monos which must be isos

David J. Fieldhouse

Department of Mathematics and Statistics, University of Guelph, Guelph N1G 2W1, Ontario, Canada

Received 21 March 1984

Copyright © 1984 David J. Fieldhouse. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Orzech [1] has shown that every surjective endomorphism of a noetherian module is an isomorphism. Here we prove analogous results for injective endomorphisms of noetherian injective modules, and the duals of these results. We prove that every injective endomorphism, with large image, of a module with the descending chain condition on large submodules is an isomorphism, which dualizes a result of Varadarajan [2]. Finally we prove the following result and its dual: if p is any radical then every surjective endomorphism of a module M, with kernel contained in pM, is an isomorphism, provided that every surjective endomorphism of pM is an isomorphism.