arXiv subjects

Muhammad Zafrullah

Publications and source records attributed to Muhammad Zafrullah.

20 records · Page 2Linked to original sources

arXiv2008

Unique representation domains, II

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 pairwise *-comaximal ideals with prime radical. When * is the t-operation we call the *-URD simply a URD. Any unique factorization domain is a URD. Generalizing and unifying results due to Zafrullah and Brewer-Heinzer, we give conditions for a *-ideal to be a unique *-product of pairwise *-comaximal ideals with prime radical and characterize *-URDs. We show that the class of URDs includes rings of Krull type, the generalized Krull domains introduced by El Baghdadi and weakly Matlis domains whose t-spectrum is treed. We also study when the property of being a URD extends to some classes of overrings, such as polynomial extensions, rings of fractions and rings obtained by the D+XD_S[X] construction.

math.AC↗

arXiv2007

Some remarks on Prüfer $\star $--multiplication domains and class groups

Let $D$ be an integral domain with quotient field $K$ and let $X$ be an indeterminate over $D$. Also, let $\boldsymbol{\mathcal{T}}:=\{T_λ\mid λ \in Λ\}$ be a defining family of quotient rings of $D$ and suppose that $\ast $ is a finite type star operation on $D$ induced by $\boldsymbol{\mathcal{T}}$. We show that $D$ is a P$ \ast $MD (resp., P$v$MD) if and only if $(\co_D(fg))^{\ast}=(\co_D(f)\co_D(g))^{\ast}$ (resp., $(\co_D(fg))^{w}=(\co_D(f)\co_D(g))^{w}$) for all $0 \ne f,g \in K[X]$. A more general version of this result is given in the semistar operation setting. We give a method for recognizing P$v$MD's which are not P$\ast $MD's for a certain finite type star operation $\ast $. We study domains $D$ for which the $\ast $--class group $\Cl^{\ast}(D)$ equals the $t$--class group $\Cl^{t}(D)$ for any finite type star operation $\ast $, and we indicate examples of P$v$MD's $D$ such that $\Cl^{\ast}(D)\subsetneq \Cl^{t}(D)$. We also compute $\Cl^v(D)$ for certain valuation domains $D$.

math.AC↗