Modules in Which Every Fully Invariant Submodule is Essential in a Direct Summand |
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| Authors: | Gary F. Birkenmeier Bruno J. Müller S. Tariq Rizvi |
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| Affiliation: | 1. Department of Mathematics , University of Louisiana at Lafayette , Lafayette, LA, 70504, USA;2. Department of Mathematics , McMaster University , Hamilton, Ontario, L85 4K1, Canada;3. Department of Mathematics , Ohio State University at Lima , Lima, OH, 45804, USA |
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| Abstract: | A module M is called extending if every submodule of M is essential in a direct summand. We call a module FI-extending if every fully invariant submodule is essential in a direct summand. Initially we develop basic properties in the general module setting. For example, in contrast to extending modules, a direct sum of FI-extending modules is FI-extending. Later we largely focus on the specific case when a ring is FI-extending (considered as a module over itself). Again, unlike the extending property, the FI-extending property is shown to carry over to matrix rings. Several results on ring direct decompositions of FI-extending rings are obtained, including a proper generalization of a result of C. Faith on the splitting-off of the maximal regular ideal in a continuous ring. |
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