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...

For a partition λ of n ∈ N , let I Sp be the ideal R = K [ x 1 … ] generated by all Specht polynomials shape . In previous paper, second author showed that if / is Cohen-Macaulay, then either ( − d ) or and converse true char 0 this we compute Hilbert series for Hence, get Castelnuovo-Mumford regularity when it Cohen-Macaulay. particular, has + 2 -linear resolution in Cohen–Macaulay case.

A basic fact about Cohen reals is that adding λ Cohen reals cannot produce more than λ of Cohen reals. More precisely, if 〈rα|α < λ〉 are λ-Cohen generic reals over V , then in V [〈rα|α < λ〉] there is no λ -Cohen generic real over V . But if instead of dealing with one universe V we consider two, then the above may no longer be true. The purpose of this paper is to produce models V1 ⊆ V2 such th...

In this paper, we use a characterization of R-modules N such that fdRN = pdRN to characterize Cohen-Macaulay rings in terms of various dimensions. This is done by setting N to be the dth local cohomology functor of R with respect to the maximal ideal where d is the Krull dimension of R.