RINGS WITH A RIGHT DUO FACTOR RING BY AN IDEAL CONTAINED IN THE CENTER

摘要

This article concerns a ring property that arises from combining one-sided duo factor rings and centers. A ring R is called right CIFD if R/I is right duo by some proper ideal I of R such that I is contained in the center of R. We first see that this property is seated between right duo and right it-duo, and not left-right symmetric. We prove, for a right CIFD ring R, that W (R) coincides with the set of all nilpotent elements of R; that R/P is a right duo domain for every minimal prime ideal P of R; that R/W (R) is strongly right bounded; and that every prime ideal of R is maximal if and only if R/W (R) is strongly regular, where W (R) is the Wedderburn radical of R. It is also proved that a ring R is commutative if and only if D3(R) is right CIFD, where D3(R) is the ring of 3 by 3 upper triangular matrices over R whose diagonals are equal. Furthermore, we show that the right CIFD property does not pass to polynomial rings, and that the polynomial ring over a ring R is right CIFD if and only if R/I is commutative by a proper ideal I of R contained in the center of R.