A note on G-type rings

Soltanpour, Somayeh; Moslemi, Bahman

doi:10.22055/jamm.2025.46283.2264

A note on G-type rings

نوع مقاله : مقاله پژوهشی

نویسندگان

Petroleum University of Technology: Ahvaz, Iran

10.22055/jamm.2025.46283.2264

چکیده

In this article, we introduce and examine the concept of $G$-type rings. A ring $R$ is classified as a $G$-type ring if its total quotient ring, denoted by $Q$, is generated by a countable number of elements over $R$ as an $R$-algebra. Formally, $Q = R[S^{-1}]$, where $S$ is a countable set of regular elements in $R$. We establish that $R$ is $G$-type if and only if there exists a countable set of regular elements, denoted as $S$, such that every prime ideal disjoint from $S$ consists solely of zero-divisors. It is shown that whenever a ring $T$ is countably generated over a subring $R$, as an $R$-algebra, and $T$ is strongly algebraic over $R$, then $R$ is $G$-type if and only if $T$ is $G$-type.

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موضوعات

ساختارهای جبری

عنوان مقاله [English]

A note on G-type rings

نویسندگان [English]

Somayeh Soltanpour
Bahman Moslemi

Petroleum University of Technology: Ahvaz, Khozestan, 2016-01-21 to present

چکیده [English]

In this article, we introduce and examine the concept of $G$-type rings. A ring $R$ is classified as a $G$-type ring if its total quotient ring, denoted by $Q$, is generated by a countable number of elements over $R$ as an $R$-algebra. Formally, $Q = R[S^{-1}]$, where $S$ is a countable set of regular elements in $R$. We establish that $R$ is $G$-type if and only if there exists a countable set of regular elements, denoted as $S$, such that every prime ideal disjoint from $S$ consists solely of zero-divisors.
It is shown that
whenever a ring $T$ is countably generated over a subring $R$, as
an $R$-algebra, and $T$ is strongly algebraic over $R$, then $R$
is $G$-type if and only if $T$ is $G$-type.

کلیدواژه‌ها [English]