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.