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Let δ be a (d-1)-dimensional simplicial complex and let h(δ)=(h0,h1,...,hd) be its h-vector. A recent result of Murai and Terai guarantees that if δ satisfies Serre's condition (Sr), then (h0,h1,...,hr) is an M-vector and hr+hr+1+...+hd is nonnegative. In this article, we extend the result of Murai and Terai by giving r extra necessary conditions. More precisely, we prove that if δ satisfies Serre's condition (Sr), then iihr+i+1ihr+1+...+i+d-rihd, 0≤i≤r≤d, are all nonnegative  

Let Γ(R) be the zero-divisor graph of a commutative ring R. An interesting question was proposed by Anderson, Frazier, Lauve, and Livingston: For which finite commutative rings R is Γ (R) planar? We give an answer to this question. More precisely, we prove that if R is a local ring with at least 33 elements, and Γ(R) ≠ 0, then Γ(R) is not planar. We use the set of the associated primes to find the minimal length of a cycle in Γ(R). Also, we determine the rings whose zero-divisor graphs are complete r-partite graphs and show that for any ring R and prime number p, p ≥ 3, if Γ(R) is a finite complete p-partite graph, then Z(R) = p2, R = p3, and R is isomorphic to exactly one of the rings ℤp3,... 

The unit graph corresponding to an associative ring R is the graph obtained by setting all the elements of R to be the vertices and defining distinct vertices x and y to be adjacent if and only if x + y is a unit of R. By a constructive method, we derive necessary and sufficient conditions for unit graphs to be Hamiltonian  

A graph is called weakly perfect if its chromatic number equals its clique number. In this note a new class of weakly perfect graphs is presented and an explicit formula for the chromatic number of such graphs is given  

A graph is called weakly perfect if its chromatic number equals its clique number. In this paper a new class of weakly perfect graphs arising from rings are presented and an explicit formula for the chromatic number of such graphs is given. Copyright  

In this note, we introduce and investigate the notion of (Formula presented.) monomial ideals. We give an explicit relation of the (Formula presented.) property to a monomial ideal and its polarization. Further, we characterize the (Formula presented.) property of the ordinary as well as the symbolic third or more powers of squarefree monomial ideals. © 2022 Taylor & Francis Group, LLC  

For a given (d-1)-dimensional simplicial complex Γ, we denote its h-vector by h(Γ)=(h 0(Γ),h 1(Γ),...,hd (Γ)) and set h -1(Γ)=0. The known Swartz equality implies that if Δ is a (d-1)-dimensional Buchsbaum simplicial complex over a field, then for every 0 ≤ i ≤ d, the inequality ihi (Δ)+(d-i+1)h i-1(Δ) ≥ 0 holds true. In this paper, by using these inequalities, we give a simple proof for a result of Terai (N. Terai, On h-vectors of Buchsbaum Stanley-Reisner rings, Hokkaido Math. J. 25(1) (1996), 137-148) on the h-vectors of Buchsbaum simplicial complexes. We then generalize the Swartz equality (E. Swartz, Lower bounds for h-vectors of k-CM, independence, and broken circuit complexes, SIAM J.... 

A classical question of Yoneda asks when the tensor product of two injective modules is injective. A complete answer to this question was given by Enochs and Jenda in 1991. In this paper the analogue question for pure-injective modules is studied  

Let K be a field and S = K[x1,...,xn] be the polynomial ring in n variables over the field K. Let G be a forest with p connected components G1,...,Gp and let I = I(G) be its edge ideal in S. Suppose that di is the diameter of Gi, 1 ≤ i ≤ p, and consider d = max{di I 1 ≤ i ≤ p}. Morey has shown that for every t ≥ 1, the quantity max is a lower bound for depth(S/It). In this paper, we show that for every t ≥ 1, the mentioned quantity is also a lower bound for sdepth(S/It). By combining this inequality with Burch's inequality, we show that any sufficiently large powers of edge ideals of forests are Stanley. Finally, we state and prove a generalization of our main theorem  

Let K be a field and S = K[x1,...,xn] be the polynomial ring in n variables over the field K. In this paper, it is shown that Stanley's conjecture holds for I and S/I if I is a product of monomial prime ideals or I is a high enough power of a polymatroidal or a stable ideal generated in a single degree  

Recently, Haghighi, Terai, Yassemi, and Zaare-Nahandi introduced the notion of a sequentially (Sr) simplicial complex. This notion gives a generalization of two properties for simplicial complexes: being sequentially Cohen-Macaulay and satisfying Serre's condition (Sr). Let Δ be a (d-1)-dimensional simplicial complex with Γ(Δ) as its algebraic shifting. Also let (hi,j(Δ))0≤j≤i≤d be the h-triangle of Δ and (hi,j(Γ(Δ)))0≤j≤i≤d be the h-triangle of Γ(Δ). In this paper, it is shown that for a Δ being sequentially (Sr) and for every i and j with 0≤j≤i≤r-1, the equality hi,j(Δ)=hi,j(Γ(Δ)) holds true  

Let Δ be a simplicial complex and χ be an s-coloring of Δ. Biermann and Van Tuyl have introduced the simplicial complex Δχ. As a corollary of Theorems 5 and 7 in their 2013 article, we obtain that the Stanley–Reisner ring of Δχ over a field is Cohen–Macaulay. In this note, we generalize this corollary by proving that the Stanley–Reisner ideal of Δχ over a field is set-theoretic complete intersection. This also generalizes a result of Macchia  

A concept of Cohen–Macaulay in codimension t is defined and characterized for arbitrary finitely generated modules and coherent sheaves by Miller, Novik, and Swartz in 2011. Soon after, Haghighi, Yassemi, and Zaare-Nahandi defined and studied CMt simplicial complexes, which is the pure version of the abovementioned concept and naturally generalizes both Cohen–Macaulay and Buchsbaum properties. The purpose of this paper is to survey briefly recent results of CMt simplicial complexes. © 2021, Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd  

In this paper, we deal with very well-covered graphs. We first describe the structure of these kinds of graphs based on the structure of Cohen–Macaulay very well-covered graphs. As an application, we analyze the structure of local cohomology of the residue rings by the edge ideals of very well-covered graphs. Also, we give different formulas of regularity and depth of these rings from known ones and we finally treat the CMt property. © 2022 Elsevier Inc  

A concept of Cohen–Macaulay in codimension t is defined and characterized for arbitrary finitely generated modules and coherent sheaves by Miller, Novik, and Swartz in 2011. Soon after, Haghighi, Yassemi, and Zaare-Nahandi defined and studied CMt simplicial complexes, which is the pure version of the abovementioned concept and naturally generalizes both Cohen–Macaulay and Buchsbaum properties. The purpose of this paper is to survey briefly recent results of CMt simplicial complexes. © 2021, Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd  

In this paper, we focus on the dimension of dual modules of local cohomology of Stanley–Reisner rings to obtain a new vector. This vector contains important information on the Serre's condition (Sr) and the CMt property as well as the depth of Stanley–Reisner rings. We prove some results in this regard including lower bounds for the depth of Stanley–Reisner rings. Further, we give a characterization of (d−1)-dimensional simplicial complexes with codimension two which are (Sd−3) but they are not Cohen–Macaulay. By using this characterization, we obtain a condition to equality of projective dimension of the Stanley–Reisner rings and the arithmetical rank of their Stanley–Reisner ideals.... 

In this paper, we characterize complete partite zero-divisor graphs of posets via the ideals of the posets. In particular, for complete bipartite zero-divisor graphs, we give a characterization based on the prime ideals of the posets  

In this paper, we discuss some recent results on graphs attached to rings. In particular, we deal with comaximal graphs, unit graphs, and total graphs. We then define the notion of cototal graph and, using this graph, we characterize the rings which are additively generated by their zero divisors. Finally, we glance at graphs attached to other algebraic structures