Related papers: Ideal depth of QF extensions

Let $k$ be a field of characteristic zero, and $R=k[x_1, \ldots, x_d]$ with $d \geq 3$ be a polynomial ring in $d$ variables. Let $\m=(x_1, \ldots, x_d)$ be the homogeneous maximal ideal of $R$. Let $\mathcal{K}$ be the kernel of the…

In this paper we explore the concept of depth of a ring extension when the overall algebra factorises as a product of two subalgebras, in particular the case of finite dimensional Hopf algebras. As a result we generalise the results by…

Let $S$ be a ring of polynomials in finitely many variables over a field. In this paper we give lower bounds for depth and Stanley depth of modules of the type $S/I^t$ for $t\geq1$, where $I$ is the edge ideal of some caterpillar and…

We prove that if $I$ is a monomial ideal with linear quotients in a ring of polynomials $S$ in $n$ indeterminates and $\operatorname{depth}(S/I)=n-2$, then $\operatorname{sdepth}(S/I)=n-2$ and, if $I$ is squarefree,…

Let $T$ be a local (Noetherian) ring and let $Q_1$ and $Q_2$ be prime ideals of $T$. We find sufficient conditions for there to exist a quasi-excellent local subring $B$ of $T$ satisfying the following conditions: (1) the completion of $B$…

We describe some basic facts about the weak subintegral closure of ideals in both the algebraic and complex-analytic settings. We focus on the analogy between results on the integral closure of ideals and modules and the weak subintegral…

Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$ denotes any field. We study grade $3$ homogeneous ideals $I \subseteq R$ defining compressed rings with socle $k(-s) \oplus k(-2s+1)$, where $s \geq3$ is some…

Let \fa be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let \cd_{\fa}(M,N) denote the supremum of the i's such that H^i_{\fa}(M,N)\neq 0. First, by using the theory of Gorenstein homological…

In this paper, we provide several new characterizations of the maximal right ring of quotients of a ring by using the relatively dense property. As a ring is embedded in its maximal right ring of quotients, we show that the endomorphism…

Dimensions like Gelfand, Krull, Goldie have an intrinsic role in the study of theory of rings and modules. They provide useful technical tools for studying their structure. In this paper we define one of the dimensions called couniserial…

Let $(A,\mathfrak{m})$ be a hypersurface ring with dimension $d$, and $M$ a MCM $A-$module with red$(M)\leq 2$ and $\mu(M)=2$ or $3$ then we have proved that depth $G(M)\geq d-\mu(M)+1$. If $e(A)=3$ and $\mu(M)=4$ then in this case we have…

The problem of computing the dimension of a left/right ideal in a group algebra F[G] of a finite group G over a field F is considered. The ideal dimension is related to the rank of a matrix originating from a regular left/right…

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring, $I$ an $\mathfrak{m}$-primary ideal of $R$ and $J$ its minimal reduction. We study the depths of $F(I)$ under certain depth assumptions on $G(I)$ and length condition on quotients of…

We study the cyclic module ${}_SR$ for a ring extension $A \| B$ with centralizer $R$ and bimodule endomorphism ring $S = End {}_BA_B$. We show that if $A \| B$ is an H-separable Hopf subalgebra, then $B$ is a normal Hopf subalgebra of $A$.…

The ring of periodic distributions on ${\mathbb{R}}^{\tt d}$ with usual addition and with convolution is considered. Via Fourier series expansions, this ring is isomorphic to the ring ${\mathcal{S}}'({\mathbb{Z}}^{\tt d})$ of all maps…

It is proved that if one of the finite modules M and N, over a local ring R, has reducible complexity and has finite Gorenstein dimension then the depth formula holds, provided TorR_i(M,N) = 0 for i>>0. We also study the vanishing of…

Given a commutative ring $R$ and finitely generated ideal $I$, one can consider the classes of $I$-adically complete, $L_0^I$-complete and derived $I$-complete complexes. Under a mild assumption on the ideal $I$ called weak pro-regularity,…

Let F. be a any free resolution of a Z^n-graded submodule of a free module over the polynomial ring K[x_1, ..., x_n]. We show that for a suitable term order on F., the initial module of the p'th syzygy module Z_p is generated by terms…

One proves that the Rees algebra of an ideal generated by three general binary forms of same degree $\geq 5$ has depth one. The proof hinges on the behavior of the Ratliff-Rush filtration for low powers of the ideal and on establishing that…

We give an upper bound for the Stanley depth of the edge ideal of a complete $k$-partite hypergraph and as an application we give an upper bound for the Stanley depth of a monomial ideal in a polynomial ring $S$. We also give a lower and an…