Related papers: Factorizations in bounded hereditary Noetherian pr…

Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ be an ideal of $R$. Suppose $M$ is a finitely generated $R$-module and $N$ is an Artinian $R$-module. We define the concept of filter coregular sequence to determine the infimum of…

Let $R$ be a Noetherian ring, $I$ and $J$ two ideals of $R$ and $t$ an integer. Let $S$ be the class of Artinian $R$-modules, or the class of all $R$-modules $N$ with $\dim_RN\leq k$, where $k$ is an integer. It is proved that $\inf\{i:…

In this paper, we say a ring $R$ is Nil$_{\ast}$-Noetherian provided that any nil ideal is finitely generated. First, we show that the Hilbert basis theorem holds for Nil$_{\ast}$-Noetherian rings, that is, $R$ is Nil$_{\ast}$-Noetherian if…

It is proved that, for a left hereditary ring, an arbitrary left module has a representation in the form of the direct sum of a stable left module and indecomposable projective left modules (if and only if an arbitrary left module has a…

We prove the following;Theorem:Let R be a prime noetherian ring with k.dimR = n, n a finite non-negative integer. We refer the reader to the definitions (1.1) of this paper.For a fixed non-negative integer m, m<n let Xm be the full set of…

We show that any homomorphism between Noetherian $F$-finite rings can be factored into a regular morphism between Noetherian $F$-finite rings followed by a surjection. This result establishes an analog of the 'smooth-by-surjective'…

Let $R$ be a commutative, Noetherian, local ring and $M$ an $R$-module. Consider the module of homomorphisms $\operatorname{Hom}_R(R/\mathfrak{a},M/\mathfrak{b} M)$ where $\mathfrak{b}\subseteq\mathfrak{a}$ are parameter ideals of $M$. When…

The class of semi-hereditary rings is an important class of rings in theories that do not assume the Noetherian condition, such as perfectoid ring theory. We prove several results concerning the structure theory of this class, focusing on…

Based upon properties of ordinal length, we introduce a new class of modules, the binary modules, and study their endomorphism ring. The nilpotent endomorphisms form a two-sided ideal, and after factoring this out, we get a commutative…

Let $A$ be a commutative noetherian ring, containing a field $k$, with $1/2\in k$, $\dim A=d$, and let $P$ be a projective $A$-module or $rank(P)=n$. In continuation of \cite{MM}, we study Homotopy obstructions for $P$ to split off a free…

Ruitenburg's Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ultimately periodic if f fixes all the generators but one. More precisely, there is N $\ge$ 0 such that f N +2 = f N , thus the period…

For a reduced Noetherian ring $R$ of characteristic $p > 0$, in this paper we discuss an extension of $R$ called its perfect closure $R^\infty$. This extension contains all $p^e$-th roots of elements of $R$, and is usually non-Noetherian.…

The class of finitely presented algebras A over a field K with a set of generators x_{1},...,x_{n} and defined by homogeneous relations of the form x_{i_1}x_{i_2}...x_{i_l}=x_{sigma(i_1)}x_{sigma(i_2)}...x_{sigma(i_l)}, where l geq 2 is a…

Let $A$ be a commutative noetherian ring, $\frak a$ be an ideal of $A$, $m,n$ be non-negative integers and let $M$ be an $A$-module such that $\Ext^i_A(A/\frak a,M)$ is finitely generated for all $i\leq m+n$. We define a class $\cS_n(\frak…

Let $\preceq$ be a preorder on a monoid $H$ and $s$ be an integer $\ge 2$. The $\preceq$-height of an $x \in H$ is the sup of the integers $k \ge 1$ for which there is a (strictly) $\preceq$-decreasing sequence $x_1,\ldots,x_k$ of…

For a certain family of complete modular lattices, we prove a Jordan--H\"older--Scheier-like" theorem with no assumptions on cardinality or well-orderedness. This family includes both lattices which are both join- and meet-continuous, as…

For a flat commutative $k$-algebra $A$ such that the enveloping algebra $A\otimes_k A$ is noetherian, given a finitely generated bimodule $M$, we show that the adic completion of the Hochschild cohomology module $HH^n(A/k,M)$ is naturally…

It is proved that a module $M$ over a Noetherian local ring $R$ of prime characteristic and positive dimension has finite flat dimension if Tor$_i^R({}^e R, M)=0$ for dim $R$ consecutive positive values of $i$ and infinitely many $e$. Here…

Our motivating goal is factorization in Krull Domains $H$ with finitely generated class group $G$. The elasticity $\rho(H)$ is the maximal number of atoms in any re-factorization of a product of $k$ atoms. The elasticities are the same as…

We study direct-sum decompositions of torsion-free, finitely generated modules over a (commutative) Bass ring $R$ through the factorization theory of the corresponding monoid $T(R)$. Results of Levy-Wiegand and Levy-Odenthal together with a…