nLab GCD ring

Redirected from "GCD rings".
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Definition

Given a commutative monoid (M,,1)(M, \cdot, 1), we say that element aMa \in M divides bMb \in M (a|ba \vert b) if there exists an element cMc \in M such that ac=ba \cdot c = b.

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

Definition

A commutative ring RR is a GCD ring if for every element aReg(R)a \in \mathrm{Reg}(R) and bReg(R)b \in \mathrm{Reg}(R), there is an element cReg(R)c \in \mathrm{Reg}(R) such that c|ac \vert a and c|bc \vert b, and for every other element dReg(R)d \in \mathrm{Reg}(R) such that d|ad \vert a and d|bd \vert b, d|cd \vert c.

See also

References

Last revised on August 19, 2024 at 15:04:53. See the history of this page for a list of all contributions to it.