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.