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Abstract
The submodules with the property of the title (N ⊆ M is strongly essential in M if ∏ I N is essential in ∏ I M for any index set I) are introduced and fully investigated.
It is shown that for each submodule N of M there exists a subset T ⊆ M such that N + T is strongly essential submodule of M and (N:T) = Ann(T), T ∩ N = 0. Basic properties of these objects and several examples are given and the counterparts of the related concepts to essential submodules are also introduced and studied. It is shown that each maximal left ideal of a left fully bounded ring is either a summand or strongly essential. Rings over which no module has a proper strongly essential submodule are characterized. It is also shown that the left Loewy rings are the only rings over which the essential submodules and strongly essential submodules of any left module coincide. Finally, a new characterization of left FBN rings is observed.
2000 Mathematics Subject Classification:
ACKNOWLEDGMENTS
We wish to thank the referees for their careful reading and, in particular, for extensive help with improving the presentation of this article. The first author would like to thank Professor G. Krause and the Department of Mathematics of the University of Manitoba for their hospitality during his six months visit to Winnipeg. The second author was in part supported by a grant from IPM (No. 84160037).
Notes
Communicated by E. R. Puczylowski