investigating modules with partial endomorphisms having μ-small kernels

In this paper, we introduce and study the concept of generalized monoform modules (g-m modules, for short) which is a proper generalization of that of monoform modules. we present some of their examples, properties and characterizations. it is shown that over a commutative ring $r$, the properties monoform, small monoform, g-m, compressible, uniform and weakly co-hopfian are all equivalent. moreover, we demonstrate that a ring r is an injective semisimple ring iff any r-module is g-m. further, we prove a similar theorem to hilbert’s basis theorem for monoform, small monoform and $g-m$ modules.