Modules whose certain submodules are essentially embedded in direct summands

Abstract

It is well known that, if the ring has acc on essential right ideals, then for every quasi-continuous module over the ring, the finite exchange property implies the full exchange property. In this paper, we obtain the former implication for the generalizations of quasi-continuous modules over a ring with acc on right annhilators of elements of the module. Moreover, we focus on direct sums and direct summands of weak $C_{12} $ modules i.e., modules with the property that every semisimple submodule can be essentially embedded in a direct summand. To this end, we prove that since weak $ C_{12} $ is closed under direct sums. Amongst other results, we provide several counterexamples including the tangent bundle of a real sphere of odd dimension over its coordinate ring for the open problem of whether weak $ C_{12}$ implies the $ C_{12} $ condition.