Let C ⊂ N be an affine semigroup, and R = K[C] its semigroup ring. This paper is a collection of various results on “C-graded” R-modules M = ⊕ c∈C Mc, especially, monomial ideals of R. For example, we show the following: If R is normal and I ⊂ R is a radicalmonomial ideal (i.e., R/I is a generalization of Stanley-Reisner rings), then the sequentially Cohen-Macaulay property of R/I is a topologi...

We introduce sequentially Sr modules over a commutative graded ring and sequentially Sr simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially Cohen-Macaulay, and satisfying Serre’s condition Sr . In analogy with the sequentially CohenMacaulay property, we show that a simplicial complex is sequentially Sr if and only if its pure i-skeleton...

The arithmetic degree, the smallest extended degree, and the homological degree are invariants that have been proposed as alternatives of the degree of a module if this module is not Cohen-Macaulay. We compare these degree functions and study their behavior when passing to the generic initial or the lexicographic submodule. This leads to various bounds and to counterexamples to a conjecture of ...

Let G be a finite simple graph on a vertex set V (G) = {x11, . . . , xn1}. Also let m1, . . . , ,mn ≥ 2 be integers and G1, . . . , Gn be connected simple graphs on the vertex sets V (Gi) = {xi1, . . . , ximi}. In this paper, we provide necessary and sufficient conditions on G1, . . . , Gn for which the graph obtained by attaching the Gi to G is unmixed or vertex decomposable. Then we character...

Let  be a local Cohen-Macaulay ring with infinite residue field,  an Cohen - Macaulay module and  an ideal of  Consider  and , respectively, the Rees Algebra and associated graded ring of , and denote by  the analytic spread of  Burch’s inequality says that  and equality holds if  is Cohen-Macaulay. Thus, in that case one can compute the depth of associated graded ring of  as  In this paper we ...

Let be a commutative Noetherian ring and let I be a proper ideal of . D’Anna and Fontana in [6] introduced a new construction of ring,  named amalgamated duplication of along I. In this paper by considering the ring homomorphism , it is shown that if , then , also it is proved that if , then there exists  such that . Using this result it is shown that if is generically Cohen-Macaulay (resp. gen...

let $(r,m)$ be a commutative noetherian local ring, $m$ a finitely generated $r$-module of dimension $d$, and let $i$ be an ideal of definition for $m$. in this paper, we extend cite[corollary 10(4)]{p} and also we show that if $m$ is a cohen-macaulay $r$-module and $d=2$, then $lambda(frac{widetilde{i^nm}}{jwidetilde{i^{n-1}m}})$ does not depend on $j$ for all $ngeq 1$, where $j$ is a minimal ...

‎We introduce a generalization of the notion of‎ depth of an ideal on a module by applying the concept of‎ local cohomology modules with respect to a pair‎ ‎of ideals‎. ‎We also introduce the concept of $(I,J)$-Cohen--Macaulay modules as a generalization of concept of Cohen--Macaulay modules‎. ‎These kind of modules are different from Cohen--Macaulay modules‎, as an example shows‎. ‎Also an art...

Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially Cohen-Macaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the Castelnuovo-Mumford regularity of R/I(G) can be determined from the invariants of G.