## 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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### References:

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