Let $M$ be a finitely generated $\mathbb{Z} $-graded module over the standard graded polynomial ring $R=K[X_1, \ldots , X_d]$ with $K$ a field, and let $H_M(t)=Q_M(t)/ (1-t)^d$ be the Hilbert series of~$M$. We introduce the Hilbert regularity of~$M$ as the lowest possible value of the Castelnuovo-Mumford regularity for an $R$-module with Hilbert series $H_M$. Our main result is an arithmetical description of this invariant which connects the Hilbert regularity of~$M$ to the smallest~$k$ such that the power series $Q_M(1-t)/(1-t)^k$ has no negative coefficients. Finally, we give an algorithm for the computation of the Hilbert regularity and the Hilbert depth of an $R$-module.

This paper proves that all power series over a maximal ordered Cauchy complete non-Archimedean field satisfy the intermediate value theorem on every closed interval. Hensel's lemma for restricted power series is the main tool of the proof.

Let $R$ be a commutative Noetherian ring of dimension~$d$ and \[ B=R[X_1,\ldots ,X_m,Y_1^{\pm 1},\ldots ,Y_n^{\pm 1}] \] a Laurent polynomial ring over $R$. If $A=B[Y,f^{-1}]$ for some $f\in R[Y]$, then we prove the following results:

(i) if $f$ is a monic polynomial, then the Serre dimension of $A$ is $\leq d$. The case $n=0$ is due to Bhatwadekar, without the condition that $f$ is a monic polynomial.

(ii) The $p$th Euler class group $E^p(A)$ of $A$, defined by Bhatwadekar and Sridharan, is trivial for $p\geq \max \{d+1,\dim A -p+3\}$. The case $m=n=0$ is due to Mandal and Parker.

Using discrete Morse theory for simplicial complexes we determine the homotopy type of ideal zero-divisor complex for finite rings and for rings with infinitely many maximal ideals.

In an integral domain $R$, a nonzero ideal is called a \textit {weakly $ES$-stable ideal} if it can be factored into a product of an invertible ideal and an idempotent ideal of $R$; and $R$ is called a \textit {weakly $ES$-stable domain} if every nonzero ideal is a weakly $ES$-stable ideal. This paper studies the notion of weakly $ES$-stability in various contexts of integral domains such as Noetherian and Mori domains, valuation and Pr\"ufer domains, pullbacks and more. In particular, we establish strong connections between this notion and well-known stability conditions, namely, Lipman, Sally-Vasconcelos and Eakin-Sathaye stabilities.

Let $(A,\mathfrak{m} )$ be a Gorenstein local ring, and let $M$ be an $A$ module of finite length and finite projective dimension. We prove that the Loewy length of $M$ is greater than or equal to the order of~$A$.

Let $A\subset B$ be an extension of commutative reduced rings and $M\subset N$ an extension of positive commutative cancellative torsion-free monoids. We prove that $A$ is subintegrally closed in $B$ and $M$ is subintegrally closed in $N$ if and only if the group of invertible $A$-submodules of $B$ is isomorphic to the group of invertible $A[M]$-submodules of $B[N]$ Theorem~\ref {6t2} (b), (d). In the case $M=N$, we prove the same without the assumption that the ring extension is reduced Theorem~\ref {6t2} (c), (d).