M. Zafrullah

activity

20072022

⋆

--multiplication domains and class groups

9 citations · 12 across the 14 of their papers we have counts for

collaborators

Showing math.ACShow all

18 papers · 1 filter

math.AC2022★ 1 cited

On Chevalley's Extension Theorem

Muhammad Zafrullah

Professor Daniel Anderson informed me, recently, that there is an error in the proof of Theorem 56 of Kaplansky's book on Commutative Rings. His (Dan's) reason was "He (Kaplansky)…

math.AC2022

Condensed domains and the $D + X L [X]$ construction

Muhammad Zafrullah

Let $D$ be an integral domain with quotient field $K$ and let $I$ be the set of nonzero ideals of $D$ . Call, for $I, J \in I (D)$ , the product $I J$ of ideal…

math.AC2022

Integral domains and the IDF property

Felix Gotti, Muhammad Zafrullah

An integral domain $D$ is called an irreducible-divisor-finite domain (IDF-domain) if every nonzero element of $D$ has finitely many irreducible divisors up to associates. The stud…

math.AC2021

Revisiting G-Dedekind domains

Muhammad Zafrullah

Let $R$ be an integral domain with $q f (R) = K$ and let $F (R)$ be the set of nonzero fractional ideals of $R .$ Call $R$ a dually compact domain (DCD) if for each $I \in F (R)$ the ideal…

math.AC2020

Domains whose ideals meet a universal restriction

Muhammad Zafrullah

Let $S (D)$ represent a set of proper nonzero ideals $I (D)$ (resp., $t$ -ideals $I_{t} (D)$ ) of an integral domain $D \neq = q f (D)$ and let $P$ be a valid property of ideals of $D .$ We…

math.AC2019

Two generalizations of Krull domains

Shiqi Xing, Daniel D. Anderson, Muhammad Zafrullah

In this paper we introduce two new generalizations of Krull domains: $*$ -almost independent rings of Krull type ( $*$ -almost IRKTs) and $*$ -almost generalized Krull domains…

math.AC2019

Almost discrete valuation domains

Daniel D. Anderson, Shiqi Xing, Muhammad Zafrullah

Let $D$ be an integral domain. Then $D$ is an almost valuation (AV-)domain if for $a, b \in D ∖ {0}$ there exists a natural number $n$ with $a^{n} ∣ b^{n}$ or $b^{n}\mid…

math.AC2019

On $*$ -homogeneous ideals

Muhammad Zafrullah

Let $*$ be a star operation of finite character. Call a $*$ -ideal $I$ of finite type a $*$ -homogeneous ideal if $I$ is contained in a unique maximal $*$ -ideal $M=M(…

math.AC2018

$t$ -local domains and valuation domains

Marco Fontana, Muhammad Zafrullah

In a valuation domain $(V, M)$ every nonzero finitely generated ideal $J$ is principal and so, in particular, $J = J^{t}$ , hence the maximal ideal $M$ is a $t$ -ideal. Therefore, the $t$ …

math.AC2018

On $*$ -Semi Homogeneous Domains

Daniel D. Anderson, Muhammad Zafrullah

Let $*$ be a finite character star operation defined on an integral domain $D .$ Call a nonzero $*$ -ideal $I$ of finite type a $*$ -homogeneous ( $*$ -homog) ideal, if…

math.AC2017

$⋆$ -super potent domains

Evan Houston, Muhammad Zafrullah

For a finite-type star operation $⋆$ on a domain $R$ , we say that $R$ is $⋆$ -super potent if each maximal $⋆$ -ideal of $R$ contains a finitely generated ideal $I$ such…

math.AC2017

On $⋆$ -Power Conductor domains

Daniel D. Anderson, Evan Houston, Muhammad Zafrullah

Let $D$ be an integral domain and $⋆$ a star operation defined on $D$ . We say that $D$ is a $⋆$ -power conductor domain ( $⋆$ -PCD) if for each pair $a,b\in D\backslash…

math.AC2017★ 1 cited

Completely integrally closed Prufer $v$ -multiplication domains

D. D. Anderson, D. F. Anderson, M. Zafrullah

We study the effects on $D$ of assuming that the power series ring $D [[X]]$ is a $v$ -domain or a PVMD. We show that a PVMD $D$ is completely integrally closed if and only if $\bigc…

math.AC2009

A " $v$ -operation free" approach to Prüfer $v$ -multiplication domains

Marco Fontana, Muhammad Zafrullah

The so called Prüfer $v$ -multiplication domains (P $v$ MD's) are usually defined as domains whose finitely generated nonzero ideals are $t$ -invertible. These domains generalize Prüfe…

math.AC2009

On $v$ -domains: a survey

Marco Fontana, Muhammad Zafrullah

An integral domain $D$ is a $v$ --domain if, for every finitely generated nonzero (fractional) ideal $F$ of $D$ , we have $(F F^{- 1})^{- 1} = D$ . The $v$ --domains generalize Prüfer and K…

math.AC2008★ 1 cited

On $v$ --domains and star operations

D. D. Anderson, David F. Anderson, Marco Fontana +1

Let $*$ be a star operation on an integral domain $D$ . Let $\f(D)$ be the set of all nonzero finitely generated fractional ideals of $D$ . Call $D$ a $*$ --Prüfer (respectively…

math.AC2008

Unique representation domains, II

Said El Baghdadi, Stefania Gabelli, Muhammad Zafrullah

Given a star operation * of finite type, we call a domain R a *-unique representation domain (*-URD) if each *-invertible *-ideal of R can be uniquely expressed as a *-product of p…

math.AC2007★ 9 cited

Some remarks on Prüfer $⋆$ --multiplication domains and class groups

David F. Anderson, Marco Fontana, Muhammad Zafrullah

Let $D$ be an integral domain with quotient field $K$ and let $X$ be an indeterminate over $D$ . Also, let $T := {T_{λ} ∣ λ \in Λ}$ be a defining family of qu…

On Chevalley's Extension Theorem

Condensed domains and the D+XL[X] construction

Integral domains and the IDF property

Revisiting G-Dedekind domains

Domains whose ideals meet a universal restriction

Two generalizations of Krull domains

Almost discrete valuation domains

On ∗-homogeneous ideals

t-local domains and valuation domains

On ∗-Semi Homogeneous Domains

⋆-super potent domains

On ⋆-Power Conductor domains

Completely integrally closed Prufer v-multiplication domains

A "v-operation free" approach to Prüfer v-multiplication domains

On v-domains: a survey

On v--domains and star operations

Unique representation domains, II

Some remarks on Prüfer ⋆--multiplication domains and class groups

Condensed domains and the $D + X L [X]$ construction

On $*$ -homogeneous ideals

$t$ -local domains and valuation domains

On $*$ -Semi Homogeneous Domains

$⋆$ -super potent domains

On $⋆$ -Power Conductor domains

Completely integrally closed Prufer $v$ -multiplication domains

A " $v$ -operation free" approach to Prüfer $v$ -multiplication domains

On $v$ -domains: a survey

On $v$ --domains and star operations

Some remarks on Prüfer $⋆$ --multiplication domains and class groups