Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. There is variety of nice results about approximately Cohen-Macaulay rings. These were done by Goto. In this paper we prove some these for modules and generalize the concept rings to modules. It seen that when $M$ an module, any proper ideal $I$ have $grade(I,M) \geq \dim_R M -\dim_R M/IM -1$. Specially $R$ itself, obtain interval $gr...

Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical counterparts, these dimensions rarely come with practical and robust criteria for finiteness, even over commutative noetherian local rings. Indeed, over such a ring ...

We already know that the noncommutative N-graded Noetherian algebras resemble commutative local rings in many respects. also have important property every minimal acyclic complex of finitely generated graded free modules is totally acyclic, and we want to generalize such properties algebra. By generalizing conclusions about combining what algebras, identify a class with acyclic. discuss how rel...