Answer & Explanation
Given an integral domain R, we say that
On the other hand, we say that a and b are comaximal if there are
It's easy to see that comaximal
As the names suggest, these aren't necessarily Bézout domains, because we only have the "Bézout relationship" for coprime elements.
But, it turns out that we can use Pre-Bézout domains to characterize Bézout domains among the class of GCD domains. More exactly, it's true the following.
Theorem: Let R be an integral domain. TFAE:
i) R is a Bézout domain.
ii) R is a GCD Pre-Bézout domain.
WLOG, we can suppose that
As R is a GCD domain, then
By an elementary property of gcds we have that
Finally, if we multiply by d the above equality we get
Thus d is a R-linear combination of a and b. Hence, R is Bézout domain.
In conclusion, according to Cohn, the class of domains you are looking for are known as Pre-Bézout domains, and these aren't necessarily Bézout domains, let alone PIDs.
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