Rings Whose Cyclic Modules Are Direct Sums of Extending Modules

Abstract

Abstract. Dedekind domains, Artinian serial rings and right uniserial rings share the following property: Every cyclic right module is a direct sum of uniform modules. We first prove the following improvement of the well-known Osofsky-Smith theorem: A cyclic module with every cyclic subfactor a direct sum of extending modules has finite Goldie dimension. So, rings with the above-mentioned property are precisely rings of the title. Non-commutative rings whose cyclic (right) modules satisfy injectivity-like properties have been studied in the last four decades since the first non-trivial result in this direction was proved by Osofsky ([16] and [17]): She showed that a ring is semisimple Artinian if its cyclic right modules are injective. Klatt and Levy characterised in [12] commutative rings whose factors are self-injective, a noncommutative analogue of which was studied by Ahsan ([1, 2]) and Koehler ([13]), where they considered rings whose cyclic right (equivalently left) modules are quasi-injective, namely qc-rings.

Rings and modules which are stable under automorphisms of their injective hulls

“…Characterizing rings via homological properties of their cyclic modules is a problem that has been studied extensively in the last fifty years. A most recent account of results related to this prototypical problem may be found in [8], and a recent addition in [2]. Another question raised in [12] is the following: What is the structure of rings whose cyclic right modules are automorphism-invariant?…”