nLab GCD ring

Definition

Given a commutative monoid $(M, \cdot, 1)$, we say that element $a \in M$ divides $b \in M$ ($a \vert b$) if there exists an element $c \in M$ such that $a \cdot c = b$.

Given a commutative ring $R$, let $\mathrm{Reg}(R)$ be the multiplicative submonoid of regular elements in $R$.

Definition

A commutative ring $R$ is a GCD ring if for every element $a \in \mathrm{Reg}(R)$ and $b \in \mathrm{Reg}(R)$, there is an element $c \in \mathrm{Reg}(R)$ such that $c \vert a$ and $c \vert b$, and for every other element $d \in \mathrm{Reg}(R)$ such that $d \vert a$ and $d \vert b$, $d \vert c$.

See also

• commutative ring

• Bézout ring

• unique factorization ring

• GCD domain

• Euclidean ring

References

• Henri Lombardi, Claude Quitté (2010): Commutative algebra: Constructive methods (Finite projective modules) Translated by Tania K. Roblo, Springer (2015) (doi:10.1007/978-94-017-9944-7, pdf)