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Definition
Given a commutative monoid , we say that element divides () if there exists an element such that and . If the commutative monoid has an absorbing element , then for all , .
Definition
A commutative ring is a GCD ring if for every element and , there is an element such that and , and for every other element such that and , .
Definition
A commutative ring is a GCD ring if there is a function such that for every element and , and , and for every other function such that and , .
Definition
A commutative ring is a GCD ring if there are functions , , and such that for every element and , and , and for every other triple of functions , , such that and , there is a function such that .
Definition
A commutative ring is a GCD ring if there are functions , , and such that for every element and , and , and is a semilattice with unit element .
The last definition implies that GCD rings are algebraic.
See also
References