arXiv subjects

Moshe Roitman

Publications and source records attributed to Moshe Roitman.

8 recordsLinked to original sources

arXiv2022

On the Gleason-Kahane-\.{Z}elazko theorem for associative algebras

The classical Gleason-Kahane-\.{Z}elazko Theorem states that a linear functional on a complex Banach algebra not vanishing on units, and such that $\Lambda(\mathbf 1)=1$, is multiplicative, that is, $\Lambda(ab)=\Lambda(a)\Lambda(b)$ for all $a,b\in A$. We study the GK\.Z property for associative unital algebras, especially for function algebras. In a GK\.Z algebra $A$ over a field of at least $3$ elements, and having an ideal of codimension $1$, every element is a finite sum of units. A real or complex algebra with just countably many maximal left (right) ideals, is a GK\.Z algebra. If $A$ is a commutative algebra, then the localisation $A_{P}$ is a GK\.Z-algebra for every prime ideal $P$ of $A$. Hence the GK\.Z property is not a local-global property. The class of GK\.Z algebras is closed under homomorphic images. If a function algebra $A\subseteq \mathbb F^{X}$ over a subfield $\mathbb F$ of $\mathbb C$, contains all the bounded functions in $\mathbb F^{X}$, then each element of $A$ is a sum of two units. If $A$ contains also a discrete function, then $A$ is a GK\.Z algebra. We prove that the algebra of periodic distributions, and the unitisation of the algebra of distributions with support in $(0,\infty)$ satisfy the GK\.Z property, while the algebra of compactly supported distributions does not.

math.RA↗

arXiv2020

On strongly primary monoids and domains

A commutative integral domain is primary if and only if it is one-dimensional and local. A domain is strongly primary if and only if it is local and each nonzero principal ideal contains a power of the maximal ideal. Hence one-dimensional local Mori domains are strongly primary. We prove among other results, that if $R$ is a domain such that the conductor $(R:\widehat R)$ vanishes, then $Λ(R)$ is finite, that is, there exists a positive integer $k$ such that each non-zero non-unit of $R$ is a product of at most $k$ irreducible elements. Using this result we obtain that every strongly primary domain is locally tame, and that a domain $R$ is globally tame if and only if $Λ(R)=\infty$. In particular, we answer Problem 38 in {P.-J. Cahen, M.~Fontana, S.~Frisch, and S.~Glaz, Open problems in commutative ring theory, Commutative Algebra, Springer 2014} in the affirmative. Many of our results are formulated for monoids.

math.AC↗

arXiv2014

On finitely stable domains

We study Archimedean and locally Archimedean stable domains. We prove that a domain is stable and one-dimensional if and only if it is finitely stable and Mori. But we give examples of Archimedean stable local domains that are not one-dimensional. We also prove that a locally Archimedean stable domain satisfies accp and that Archimedean stable semilocal domains are locally Archimedean. But generally, neither Archimedean stable domains, nor Archimedean semilocal domains are necessarily locally Archimedean.

math.AC↗

arXiv2011

The Kaplansky condition and rings of almost stable range 1

We present some variants of the Kaplansky condition for a K-Hermite ring $R$ to be an elementary divisor ring; for example, a commutative K-Hermite ring $R$ is an EDR iff for any elements $x,y,z\in R$ such that $(x,y)=(1)$, there exists an element $λ\in R$ such that $x+λy=uv$, where $(u,z)=(v,1-z)=(1)$. We present an example of a a Bézout domain that is an elementary divisor ring, but it does not have almost stable range 1, thus answering a question of Warren Wm. McGovern.

math.AC↗

arXiv2004

Well-centered overrings of an integral domain

Let A be an integral domain with field of fractions K. We investigate the structure of the overrings B of A (contained in K) that are well-centered on A in the sense that each principal ideal of B is generated by an element of A. We consider the relation of well-centeredness to the properties of flatness, localization and sublocalization for B over A. If B = A[b] is a simple extension of A, we prove that B is a localization of A if and only if B is flat and well-centered over A. If the integral closure of A is a Krull domain, in particular, if A is Noetherian, we prove that every finitely generated flat well-centered overring of A is a localization of A. We present examples of (non-finitely generated) flat well-centered overrings of a Dedekind domain that are not localizations.

math.AC↗

arXiv2003

The content of a Gaussian polynomial is invertible

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 that if c(f) is an invertible ideal, then f is Gaussian. In this note we prove the converse.

math.AC↗

arXiv2003

Maximal divisorial ideals and t-maximal ideals

We give conditions for a maximal divisorial ideal to be t-maximal and show with examples that, even in a completely integrally closed domain, maximal divisorial ideals need not be t-maximal.

math.AC↗

arXiv2003

Complete integral closure and strongly divisorial prime ideals

It is well known that a domain without proper strongly divisorial ideals is completely integrally closed. In this paper we show that a domain without {\em prime} strongly divisorial ideals is not necessarily completely integrally closed, although this property holds under some additional assumptions.

math.AC↗