Extending modules which are direct sums of injective modules and semisimple modules
C Santa-Clara, PF Smith - Communications in Algebra, 1996 - Taylor & Francis
C Santa-Clara, PF Smith
Communications in Algebra, 1996•Taylor & Francis… Natural examples of extending modules are semisimple modules and injective modules.
We characterize when the direct sum of an injective module and a semisimple module is
extending, in case the ring we are working wer is right Noetherian. We also characterize the
right Goldie finite (or right nonsingular) rings R with the property that every direct sum of an
injective R-module and a semisimple R-module is extending. It remains an open problem to
characterize rings with the above property, in the general case. …
We characterize when the direct sum of an injective module and a semisimple module is
extending, in case the ring we are working wer is right Noetherian. We also characterize the
right Goldie finite (or right nonsingular) rings R with the property that every direct sum of an
injective R-module and a semisimple R-module is extending. It remains an open problem to
characterize rings with the above property, in the general case. …
It is proved that, if R is a right Noetherian ringM 1 is an injective right R-module and M 2 is a semisimple right R-module, then the right R-module M 1 + M 2 is extending if and only if M 2 is (M 1/Soc(M 1))-injective.
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