When are associates unit multiples? (Q1768057)

Let \(R\) be a commutative ring with identity. Two elements \(a,b\in R\) are called: (i) associates, denoted \(a\sim b\), if \(a| b\) and \(b| a\); (ii) strong associates, denoted \(a\approx b\), if \(a=ub\) for some unit \(u\) of \(R\); (iii) very strong associates, denoted \(a\cong b\), if \(a\sim b\) and the conditions \(a\neq 0\) and \(a=rb\) imply that \(r\) is a unit. One always has \(a\cong b\Rightarrow a\approx b\Rightarrow a\sim b\). The ring \(R\) is called: (i) strongly associate if \(a,b\in R\), \(a\sim b\) implies \(a\approx b\); (ii) présimplifiable if \(a,b\in R\), \(a\sim b\) (or \(a\approx b\)) implies \(a\cong b\). It is known that every présimplifiable ring is strongly associate. The authors establish various properties of these kinds of rings.