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

Yıl 2022, Cilt: 32 Sayı: 32, 86 - 90, 16.07.2022

Gyu Whan Chang Jun Seok Oh

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

Öz

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.

Anahtar Kelimeler

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

Kaynakça

• D.D. Anderson and M. Roitman, A characterization of cancellation ideals, Proc. Amer. Math. Soc., 125 (1997), 2853-2854.
• S. Chaopraknoi, K. Savettaseranee and P. Lertwichitsilp, Some cancellation ideal rings, Gen. Math., 13 (2005), 39-46.
• A. Geroldinger and Q. Zhong, Factorization theory in commutative monoids, Semigroup Forum, 100 (2020), 22-51.
• R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, Inc., New York, 1972.
• R. Gilmer, Commutative Semigroup Rings, Univ. Chicago Press, Chicago, 1984.