Related papers: Domains with invertible-radical factorization

In algebra, atomicity is the study of divisibility by and factorizations into atoms (also called irreducibles). In one side of the spectrum of atomicity we find the antimatter algebraic structures, inside which there are no atoms and,…

An integral domain $D$ is called an SP-domain if every ideal is a product of radical ideals. Such domains are always almost Dedekind domains, but not every almost Dedekind domain is an SP-domain. The SP-rank of $D$ provides a natural…

An ideal is a classical object of study in the field of algebraic number theory. In maximal quadratic orders of number fields, ideals usually represented by the $\mathbb Z$-basis. This form of representation is used in most of the…

In this paper we describe three different variations of prime ideals: strongly irreducible ideals, strongly prime ideals and insulated prime ideals in the context of Leavitt path algebras. We give necessary and sufficient conditions under…

Let $R$ be an infinite Dedekind domain with at most finitely many units, and let $K$ denote its field of fractions. We prove the following statement. If $L/K$ is a finite Galois extension of fields and $\mathcal{O}$ is the integral closure…

$\DeclareMathOperator{\Int}{Int}\DeclareMathOperator{\IntR}{Int{}^\text{R}}$For a domain $D$, the ring $\Int(D)$ of integer-valued polynomials over $D$ is atomic if $D$ satisfies the ascending chain condition on principal ideals. However,…

Let $S(D)$ represent a set of proper nonzero ideals $I(D)$ (resp., $t$ -ideals $I_{t}(D)$) of an integral domain $D\neq qf(D)$ and let $P$ be a valid property of ideals of $D.$ We say $S(D)$ meets $P$ (denoted $ S(D)\vartriangleleft P)$ if…

We study the integral domains D satisfying the following condition: whenever I >AB with I,A,B nonzero ideals, there exist ideals A'>A and B'>B such that I=A'B'.

The *reciprocal complement* $R(D)$ of an integral domain $D$ is the subring of its fraction field generated by the reciprocals of its nonzero elements. Many properties of $R(D)$ are determined when $D$ is a polynomial ring in $n\geq 2$…

We study a new class of infinite dimensional Lie algebras, which has important applications to the theory of integrable equations. The construction of these algebras is very similar to the one for automorphic functions and this motivates…

In this article, we study the monoid of fractional ideals and the ideal class semigroup of an arbitrary given one dimensional normal domain O obtained by an infinite integral extension of a Dedekind domain. We introduce a notion of "upper…

An integral domain $D,$ with quotient field $K,$ is a $v$-domain if for each nonzero finitely generated ideal $A$ of $D$ we have $(AA^{-1})^{-1}=D.$ It is well known that if $D$ is a $v$-domain$,$ then some quotient ring $D_{S}$ of $D$ may…

This paper surveys and develops links between polynomial invariants of finite groups, factorization theory of Krull domains, and product-one sequences over finite groups. The goal is to gain a better understanding of the multiplicative…

Using the tools of reverse mathematics in second-order arithmetic, as developed by Friedman, Simpson, and others, we determine the axioms necessary to develop various topics in commutative ring theory. Our main contributions to the field…

Let $(A,\mathfrak{m})$ be an excellent normal domain of dimension two containing a field $k \cong A/\mathfrak{m}$. An $\mathfrak{m}$-primary ideal $I$ to be a $p_g$-ideal if the Rees algebra $A[It]$ is a Cohen-Macaulay normal domain. If $k$…

Let R be an integral domain and let f(X) be a nonzero polynomial in R[X]. The content of f is the ideal c(f) generated by the coefficients of f. The polynomial f(X) is called Gaussian if c(fg)=c(f)c(g) for all g(X) in R[X]. It is well known…

The notion of PRINC domain was introduced by Salce and Zanardo (2014), motivated by the investigation of the products of idempotent matrices with entries in a commutative domain. An integral domain R is a PRINC domain if every two-generated…

We characterize the commutative rings whose ideals (resp. regular ideals) are products of radical ideals.

Using polynomial evaluation, we give some useful criteria to answer questions about divisibility of polynomials. This allows us to develop interesting results concerning the prime elements in the domain of coefficients. In particular, it is…

Recursive domain equations have natural solutions. In particular there are domains defined by strictly positive induction. The class of countably based domains gives a computability theory for possibly non-countably based topological…