On the set of non-zero elements in an integral domain whose generating principal ideal is of a special kind
Let R be an integral domain. Consider the set
S:={a∈R∖{0}:Ra+Rx is a principal ideal ∀x∈R}.
Is S a saturated multiplicative closed subset of R? If in general S is not saturated or multiplicative closed, what if we assume R is a GCD domain? Is the claim true then ?
If we assume R is local , let x∈R , then a∈S⟹∃d∈R such that Ra+Rx=Rd , where d|a,d|x , then let a′=a/d,x′=x/d , then Ra′+Rx′=R , but R is a local ring , hence one of a′ and x′ must be a unit i.e. either a|x or x|a . Thus in a local ring R ,
{a∈R∖{0}:Ra+Rx is a principal ideal ∀x∈R}
={a∈R∖{0}: for every x∈R , either x|a or a|x} ; and I can show that in any integral
domain , {a∈R∖{0}: for every x∈R , either x|a or a|x} is a saturated multiplicative closed set .
I have no idea what happens if the ring is not local .
S:={a∈R∖{0}:Ra+Rx is a principal ideal ∀x∈R}.
Is S a saturated multiplicative closed subset of R? If in general S is not saturated or multiplicative closed, what if we assume R is a GCD domain? Is the claim true then ?
If we assume R is local , let x∈R , then a∈S⟹∃d∈R such that Ra+Rx=Rd , where d|a,d|x , then let a′=a/d,x′=x/d , then Ra′+Rx′=R , but R is a local ring , hence one of a′ and x′ must be a unit i.e. either a|x or x|a . Thus in a local ring R ,
{a∈R∖{0}:Ra+Rx is a principal ideal ∀x∈R}
={a∈R∖{0}: for every x∈R , either x|a or a|x} ; and I can show that in any integral
domain , {a∈R∖{0}: for every x∈R , either x|a or a|x} is a saturated multiplicative closed set .
I have no idea what happens if the ring is not local .
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