When Some Complement of a Submodule Is a Summand

Abstract

A module M is said to satisfy the C ₁₁ condition if every submodule of M has a (i.e., at least one) complement which is a direct summand. It is known that the C ₁ condition implies the C ₁₁ condition and that the class of C ₁₁-modules is closed under direct sums but not under direct summands. We show that if M = M ₁ ⊕ M ₂, where M has C ₁₁ and M ₁ is a fully invariant submodule of M, then both M ₁ and M ₂ are C ₁₁-modules. Moreover, the C ₁₁ condition is shown to be closed under formation of the ring of column finite matrices of size Γ, the ring of m-by-m upper triangular matrices and right essential overrings. For a module M, we also show that all essential extensions of M satisfying C ₁₁ are essential extensions of C ₁₁-modules constructed from M and certain subsets of idempotent elements of the ring of endomorphisms of the injective hull of M. Finally, we prove that if M is a C ₁₁-module, then so is its rational hull. Examples are provided to illustrate and delimit the theory.

Key Words:

• C ₁₁-ring
• C ₁₁-module
• Extending module
• FI-extending module

2000 Mathematics Subject Classification:

• Primary 16D70
• Secondary 16D50, 16D40

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Addendum and Corrigendum

ACKNOWLEDGMENTS

The authors appreciate the careful reading and comments by the referee. This work was made possible by TUBITAK (NATO-D PROGRAM) and was carried out during visits of the first author to Hacettepe University in July 2003 and August 2004. The authors wish to thank both organizations for their support. The first author is grateful for the hospitality shown to him by the faculty and staff of Hacettepe University.

Notes

Communicated by T. Albu.