When does a quotient ring of a PID have the cancellation property?

Year 2022, Volume: 32 Issue: 32, 86 - 90, 16.07.2022

Gyu Whan Chang Jun Seok Oh

https://doi.org/10.24330/ieja.1102363

Abstract

An ideal $I$ of a commutative ring is called a cancellation ideal
if $IB = IC$ implies $B = C$ for all ideals $B$ and $C$.
Let $D$ be a principal ideal domain (PID), $a, b \in D$ be nonzero elements with $a \nmid b$,
$(a, b)D = dD$ for some $d \in D$, $D_a = D/aD$ be the quotient ring of $D$ modulo $aD$,
and $bD_a = (a,b)D/aD$; so $bD_a$ is a nonzero commutative ring. In this paper, we show that
the following three properties are equivalent:
(i) $\frac{a}{d}$ is a prime element and $a \nmid d^{2}$,
(ii) every nonzero ideal of $bD_a$ is a cancellation ideal,
and (iii) $bD_a$ is a field.

Keywords

Principal ideal domain, cancellation ideal, Principal ideal domain, cancellation ideal

References

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