For every finite Abelian group Γ and for all $g,a₁,...,a_k\in \Gamma$, if there exists a solution of the equation ${\sum }_{i=1}^k{a}_i{x}_i=g$ in non-negative integers $x_i\le b_i$, where $b_i$ are positive integers, then the number of such solutions is estimated from below in the best possible way.

We say that an ideal I on ${}_{\kappa }\lambda$ is semiproper if the corresponding poset ${\mathbb{P}}_I$ is semiproper. In this paper we investigate properties of semiproper ideals on ${}_{\kappa }\lambda$.

Let $R$ be a commutative ring with nonzero identity, let $ℐ\left(ℛ\right)$ be the set of all ideals of $R$ and $\delta :ℐ\left(ℛ\right)\to ℐ\left(ℛ\right)$ an expansion of ideals of $R$ defined by $I\mapsto \delta \left(I\right)$. We introduce the concept of $(\delta ,2)$-primary ideals in commutative rings. A proper ideal $I$ of $R$ is called a $(\delta ,2)$-primary ideal if whenever $a,b\in R$ and $ab\in I$, then $a^2\in I$ or $b^2\in \delta \left(I\right)$. Our purpose is to extend the concept of $2$-ideals to $(\delta ,2)$-primary ideals of commutative rings. Then we investigate the basic properties of $(\delta ,2)$-primary ideals and also discuss the relations among $(\delta ,2)$-primary, $\delta$-primary...

Let $F$ be a finite field of characteristic $p$ and $K$ a field which contains a primitive $p$th root of unity and $\mathrm{char}K\ne p$. Suppose that a classical group $G$ acts on the $F$-vector space $V$. Then it can induce the actions on the vector space $V\oplus V$ and on the group algebra $K[V\oplus V]$, respectively. In this paper we determine the structure of $G$-invariant ideals of the group algebra $K[V\oplus V]$, and establish the relationship between the invariant ideals of $K\left[V\right]$ and the vector invariant ideals of $K[V\oplus V]$, if $G$ is a unitary group or orthogonal...

The concept of $\alpha$-ideals in posets is introduced. Several properties of $\alpha$-ideals in $0$-distributive posets are studied. Characterization of prime ideals to be $\alpha$-ideals in $0$-distributive posets is obtained in terms of minimality of ideals. Further, it is proved that if a prime ideal $I$ of a $0$-distributive poset is non-dense, then $I$ is an $\alpha$-ideal. Moreover, it is shown that the set of all $\alpha$-ideals $\alpha \mathrm{Id}\left(P\right)$ of a poset $P$ with $0$ forms a complete lattice. A result analogous to separation theorem for...

When $S$ is a polynomial ring or more generally a standard graded algebra over a field $K$, with homogeneous maximal ideal $𝔪$, it is known that for an ideal $I$ of $S$, the regularity of powers of $I$ becomes eventually a linear function, i.e., $\mathrm{reg}\left(I^m\right)=dm+e$ for $m\gg 0$ and some integers $d$, $e$. This motivates writing $\mathrm{reg}\left(I^m\right)=dm+e_m$ for every $m\ge 0$. The sequence $e_m$, called the of the ideal $I$, is the subject of much research and its nature is still widely unexplored. We know that $e_m$ is eventually constant. In this article, after proving...

Let $U$ be the two-parameter quantized enveloping algebra $U_{r,s}\left({\mathrm{𝔰𝔩}}_2\right)$ and $F\left(U\right)$ the locally finite subalgebra of $U$ under the adjoint action. The aim of this paper is to determine some ring-theoretical properties of $F\left(U\right)$ in the case when $r{s}^{-1}$ is not a root of unity. Then we describe the annihilator ideals of finite dimensional simple modules of $U$ by generators.

Let $𝔞\subseteq ℂ[x_1,...,x_n]$ be a monomial ideal and $𝒥\left({𝔞}^c\right)$ the multiplier ideal of $𝔞$ with coefficient $c$. Then $𝒥\left({𝔞}^c\right)$ is also a monomial ideal of $ℂ[x_1,...,x_n]$, and the equality $𝒥\left({𝔞}^c\right)=𝔞$ implies that $0<c<n+1$. We mainly discuss the problem when $𝒥\left(𝔞\right)=𝔞$ or $𝒥\left({𝔞}^{n+1-\epsilon }\right)=𝔞$ for all $0<\epsilon <1$. It is proved that if $𝒥\left(𝔞\right)=𝔞$ then $𝔞$ is principal, and if $𝒥\left({𝔞}^{n+1-\epsilon }\right)=𝔞$ holds for all $0<\epsilon <1$ then $𝔞=(x_1,...,x_n)$. One global result is also obtained. Let $\widetilde{𝔞}$ be the ideal sheaf on ${\mathbb{P}}^{n-1}$ associated with $𝔞$. Then it is proved that the equality $𝒥\left(\widetilde{𝔞}\right)=\widetilde{𝔞}$ implies that $\widetilde{𝔞}$ is principal.

It is well-known that a probability measure $\mu$ on the circle $𝕋$ satisfies $\parallel {\mu }^n{*f-\int f\mathrm{d}m\parallel }_p\to 0$ for every $f\in L_p$, every (some) $p\in [1,\infty )$, if and only if $|\widehat{\mu }\left(n\right)|lt;1$ for every non-zero $n\in \mathbb{Z}$ ($\mu$ is strictly aperiodic). In this paper we study the a.e. convergence of ${\mu }^n*f$ for every $f\in L_p$ whenever $pgt;1$. We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of $\mu$, for the strong sweeping out property (existence of a Borel set $B$ with $lim\; sup{\mu }^n*1_B=1$ a.e. and $lim\; inf{\mu }^n*1_B=0$ a.e.). The results are extended to general compact Abelian groups...

Let $I$ be an ideal in a commutative Noetherian ring $R$. Then the ideal $I$ has the strong persistence property if and only if $\left(I^{k+1}{:}_{R}I\right)=I^k$ for all $k$, and $I$ has the symbolic strong persistence property if and only if $\left(I^{(k+1)}{:}_R{I}^{\left(1\right)}\right)=I^{\left(k\right)}$ for all $k$, where $I^{\left(k\right)}$ denotes the $k$th symbolic power of $I$. We study the strong persistence property for some classes of monomial ideals. In particular, we present a family of primary monomial ideals failing the strong persistence property. Finally, we show that every square-free monomial...