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

Authors : Gyu Whan CHANG, Jun Seok OH

Pages : 86-90

Doi:10.24330/ieja.1102363

View : 10 | Download : 8

Publication Date : 2022-07-16

Article Type : Research Paper

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 insert ignore into journalissuearticles values(PID);, $a, b \in D$ be nonzero elements with $a \nmid b$, $insert ignore into journalissuearticles values(a, b);D = dD$ for some $d \in D$, $D_a = D/aD$ be the quotient ring of $D$ modulo $aD$, and $bD_a = insert ignore into journalissuearticles values(a,b);D/aD$; so $bD_a$ is a nonzero commutative ring. In this paper, we show that the following three properties are equivalent: insert ignore into journalissuearticles values(i); $\frac{a}{d}$ is a prime element and $a \nmid d^{2}$, insert ignore into journalissuearticles values(ii); every nonzero ideal of $bD_a$ is a cancellation ideal, and insert ignore into journalissuearticles values(iii); $bD_a$ is a field.
Keywords : Principal ideal domain, cancellation ideal, Principal ideal domain, cancellation ideal