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When are associates unit multiples? (English) Zbl 1092.13002
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.

MSC:
13A05 Divisibility and factorizations in commutative rings
13A15 Ideals and multiplicative ideal theory in commutative rings
13F99 Arithmetic rings and other special commutative rings
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