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...

We introduce a new family of simple graphs, so called, growing graphs. investigate ways to modify given graph G combinatorially obtain graph. One may infinitely many graphs from single show that obtained any is Cohen–Macaulay and every chordal also prove under certain conditions, if only its clique complex grafted give several equivalent conditions in this case. Our work inspired by generalizes...