Let $𝒵\left(ℛ\right)$ be the set of zero divisor elements of a commutative ring $R$ with identity and $ℳ$ be the space of minimal prime ideals of $R$ with Zariski topology. An ideal $I$ of $R$ is called strongly dense ideal or briefly $sd$-ideal if $I\subseteq 𝒵\left(ℛ\right)$ and $I$ is contained in no minimal prime ideal. We denote by $R_K\left(ℳ\right)$, the set of all $a\in R$ for which $\overline{D\left(a\right)}=\overline{ℳ\setminus V\left(a\right)}$ is compact. We show that $R$ has property $\left(A\right)$ and $ℳ$ is compact if and only if $R$ has no $sd$-ideal. It is proved that $R_K\left(ℳ\right)$ is an essential ideal (resp., $sd$-ideal) if and only if $ℳ$ is an almost locally compact...

A fairly old problem in modular representation theory is to determine the vanishing behavior of the $\mathrm{H}om$ groups and higher $\mathrm{E}xt$ groups of Weyl modules and to compute the dimension of the $\mathbb{Z}/\left(p\right)$-vector space ${\mathrm{H}om}_{{\overline{A}}_r}({\overline{K}}_{\lambda },{\overline{K}}_{\mu })$ for any partitions $\lambda$, $\mu$ of $r$, which is the intertwining number. K. Akin, D. A. Buchsbaum, and D. Flores solved this problem in the cases of partitions of length two and three. In this paper, we describe the vanishing behavior of the groups ${\mathrm{H}om}_{{\overline{A}}_r}({\overline{K}}_{\lambda },{\overline{K}}_{\mu })$ and provide a new formula for the intertwining number for any...

The vertex algebra ${𝒲}_{1+\infty ,c}$ with central charge $c$ may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer $n\ge 1$, it was conjectured in the physics literature that ${𝒲}_{1+\infty ,-n}$ should have a minimal strong generating set consisting of $n^2+2n$ elements. Using a free field realization of ${𝒲}_{1+\infty ,-n}$ due to Kac–Radul, together with a deformed version of Weyl’s first and second fundamental theorems of invariant theory for the standard representation of ${\text{GL}}_n$,...

Quasi-injective modules over valuation domains are classified by means of complete sets of cardinal invariants.

Let $𝔽$ be any field of characteristic $p$. It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_1,V_2,...,V_p$ of $C_p$ defined over $𝔽$. Thus if $V$ is any finite dimensional $C_p$-representation there are non-negative integers $0\le n_1,n_2,...,n_k\le p-1$ such that $V\cong {\oplus }_{i=1}^k{V}_{n_i+1}$. It is also well-known there is a unique (up to equivalence) $d+1$ dimensional irreducible complex representation of $S{L}_2\left(ℂ\right)$ given by its action on the space $R_d$ of $d$ forms. Here we prove a conjecture, made by R. J. Shank, which reduces the computation of the ring...

We investigate the invariant rings of two classes of finite groups $G\le \mathrm{GL}(n,F_q)$ which are generated by a number of generalized transvections with an invariant subspace $H$ over a finite field $F_q$ in the modular case. We name these groups generalized transvection groups. One class is concerned with a given invariant subspace which involves roots of unity. Constructing quotient groups and tensors, we deduce the invariant rings and study their Cohen-Macaulay and Gorenstein properties. The other is concerned with...

We study the minimal number of elements of maximal order occurring in a zero-sumfree sequence over a finite Abelian p-group. For this purpose, and in the general context of finite Abelian groups, we introduce a new number, for which lower and upper bounds are proved in the case of finite Abelian p-groups. Among other consequences, our method implies that, if we denote by exp(G) the exponent of the finite Abelian p-group G considered, every zero-sumfree sequence S with maximal possible length over...