Related papers: Reverse mathematics of rings
In this survey article we outline the history of the twin theories of weak normality and seminormality for commutative rings and algebraic varieties with an emphasis on the recent developments in these theories over the past fifteen years.…
We survey results on factorizations of non zero-divisors into atoms (irreducible elements) in noncommutative rings. The point of view in this survey is motivated by the commutative theory of non-unique factorizations. Topics covered include…
Absolute integral closures of general commutative unital rings are explored. All rings admit absolute integral closures, but in general they are not unique. Among the reduced rings with finitely many minimal prime ideals, finite products of…
Our starting point is Mumford's conjecture, on representations of Chevalley groups over fields, as it is phrased in the preface of "Geometric Invariant Theory". After extending the conjecture appropriately, we show that it holds over an…
We say that a commutative ring R satisfies the restricted minimum (RM) condition if for all essential ideals I in R, factor R/I is an Artinian ring. We will focus on Noetherian reduced rings because in this setting known results for RM…
We introduce a weak division-like property for noncommutative rings: a nontrivial ring is fadelian if for all nonzero $a,x$ there exist $b,c$ such that $x=ab+ca$. We prove properties of fadelian rings, and construct examples of such rings…
The aim of Reverse Mathematics(RM for short)is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. These minimal axioms are almost always equivalent to the theorem, working over the base theory of RM, a weak…
In this paper, we study the classes of rings in which every proper (regular) ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals. More precisely, we investigate the stability of these properties…
Weak proregularity of an ideal in a commutative ring is a subtle generalization of the noetherian property of the ring. Weak proregularity is of special importance for the study of derived completion, and it occurs quite often in…
Let $R$ be a ring and $S$ a multiplicative subset of $R$. We introduce and study the notions of ($u$-)$S$-$w$-Noetherian modules and ($u$-)$S$-$w$-principal ideal modules. Some characterizations of these new concepts are given.
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…
Generalizing the concept of the Macaulay inverse system, we introduce a way to describe localizations of an ideal in a polynomial ring. This leads to an approach to the differential primary decomposition as a description of the affine…
The target of the present work is to give a new insight in the theory of {\it strongly weakly nil-clean} rings, recently defined by Kosan and Zhou in the Front. Math. China (2016) and further explored in detail by Chen-Sheibani in the J.…
In this paper, we introduce the concept of n-semiprimary ideals, n-powerful ideals, and n-powerful semiprimary ideals of commutative rings. We study these concepts and relate them to several generalizations of pseudo-valuation domains.
A well-known result of K\"{o}the and Cohen-Kaplansky states that a commutative ring $R$ has the property that every $R$-module is a direct sum of cyclic modules if and only if $R$ is an Artinian principal ideal ring. This motivated us to…
We provide an irreducibility test and factoring algorithm (with some qualifications) for formal power series in the unique factorization domain $R[[X]]$, where $R$ is any principal ideal domain. We also classify all integral domains arising…
In this paper, we extend the characterization of $\mathbb{Z}[x]/\ < f \ >$, where $f \in \mathbb{Z}[x]$ to be a free $\mathbb{Z}$-module to multivariate polynomial rings over any commutative Noetherian ring, $A$. The characterization allows…
We introduce a complete radical formula for modules over non-commutative rings which is the equivalence of a radical formula in the setting of modules defined over commutative rings. This gives a general frame work through which known…
In a paper on the taxonomy of 2-primal rings, examples of various types of rings that are related to commutativity such as reduced, symmetric, duo, reversible and PS~I were given in order to show that the ring class inclusions were strict.…
Students studying the Lasker-Noether theorem on primary decomposition of ideals may want to see an example of an ideal (necessarily in a non-Noetherian ring) which does not have a primary decomposition. The most well-known counterexample is…