Related papers: On the complete intersection conjecture of Murthy

Let $S = K[x_1,..., x_n]$ be a polynomial ring over a field $K$. Let $I(G) \subseteq S$ denote the edge ideal of a graph $G$. We show that the $\ell$th symbolic power $I(G)^{(\ell)}$ is a Cohen-Macaulay ideal (i.e., $S/I(G)^{(\ell)}$ is…

We prove the following result. Let k be an infinite perfect field of positive characteristic and assume that strong resolution of singularities holds over k. Let R be a localization of a commutative d-dimensional k-algebra of finite type…

Let $R = k[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $k$ and let $I$ be a monomial ideal of $R$. In this paper, we study almost Cohen-Macaulay simplicial complex. Moreover, we characterize the almost…

Let $D$ be an integrally closed domain with quotient field $K$. Let $A$ be a torsion-free $D$-algebra that is finitely generated as a $D$-module. For every $a$ in $A$ we consider its minimal polynomial $\mu_a(X)\in D[X]$, i.e. the monic…

In this paper, new algebraic and topological results on purely-prime ideals of a commutative ring (pure spectrum) are obtained. Especially, Grothendieck type theorem is obtained which states that there is a canonical correspondence between…

We investigate, using the notion of linear quotients, significative classes of connected graphs whose monomial edge ideals, not necessarily squarefree, have linear resolution, in order to compute standard algebraic invariants of the…

Given a 0-dimensional affine K-algebra R=K[x_1,...,x_n]/I, where I is an ideal in a polynomial ring K[x_1,...,x_n] over a field K, or, equivalently, given a 0-dimensional affine scheme, we construct effective algorithms for checking whether…

We study rings with infinitely (only finitely) many maximal subrings. We prove that if $M$ is a maximal left/right ideal of a ring $T$ which is not an ideal of $T$, and $R$ is the idealizer of $M$, then $T$ has at least $|R/M|+1$ maximal…

Let $I$ be an equigenerated squarefree monomial ideal in the polynomial ring $\mathbb{K}[x_1,\ldots,x_n]$, and let $\mathcal{H}$ be a uniform clutter on the vertex set $\{x_1,\ldots,x_n\}$ such that $I=I(\mathcal{H})$ is its edge ideal. A…

Throughout this abstruct $A$ will denote a noetherian commutative ring of dimension $n$. The paper has two parts. Among the interesting results in Part-1 are the following: 1) {\it suppose that $f_1, f_2, ..., f_r$ (with $r \leq n$) is a…

We classify all unmixed monomial ideals I of codimension 2 which are generically a complete intersection and which have the property that the symbolic power algebra A(I) is standard graded. We give a lower bound for the highest degree of a…

We investigate the standard graded $k$-algebras over a field $k$ of characteristic zero for which general linear forms are exact zero divisors. We formulate a conjecture regarding the Hilbert function of such rings. We prove our conjecture…

A quasi-complete intersection (q.c.i.) ideal of a local ring is an ideal with "free exterior Koszul homology"; the definition can also be understood in terms of vanishing of Andr\'e-Quillen homology functors. Principal q.c.i. ideals are…

Let $d_1,...,d_r$ be positive integers and let $I = (F_1,...,F_r)$ be an ideal generated by general forms of degrees $d_1,...,d_r$, respectively, in a polynomial ring $R$ with $n$ variables. When all the degrees are the same we give a…

Let K be a field and let S = K[x_1, ..., x_n] be a polynomial ring. Consider a homogenous ideal I in S. Let t_i denote reg(Tor_i (S/I, K)), the maximal degree of an ith syzygy of S/I. We prove bounds on the numbers t_i for i > n/2 purely in…

Let R=k[x_1,...,x_n] be a polynomial ring over a field k. Let J={j_1,...,j_t} be a subset of [n]={1,...,n}, and let m_J denote the ideal (x_{j_1},...,x_{j_t}) of R. Given subsets J_1,...,J_s of [n] and positive integers a_1,...,a_s, we…

Over an arbitrary field $\mathbb{F}$, Harbourne conjectured that $$I^{(N (r-1)+1)} \subseteq I^r$$ for all $r>0$ and all homogeneous ideals $I$ in $S = \mathbb{F} [\mathbb{P}^N] = \mathbb{F} [x_0, \ldots, x_N]$. The conjecture has been…

Let $R = k[x_1, \dotsc , x_n]$ denote the standard graded polynomial ring over a field $k$. We study certain classes of equigenerated monomial ideals with the property that the so-called complementary ideal has no linear relations on the…

Let $k$ be a field with characteristic zero, $R$ be the ring $k[x_1, \cdots, x_n]$ and $I$ be a monomial ideal of $R$. We study the Artinian local algebra $R/I$ when considered as an $R$-module $M$. We show that the largest reduced…

Let $R$ be a commutative ring with an identity different from zero and $n$ be a positive integer. Anderson and Badawi, in their paper on $n$-absorbing ideals, define a proper ideal $I$ of a commutative ring $R$ to be an $n$-absorbing ideal…