Rings with a right duo factor ring by an ideal contained in the center
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This article concerns a ring property that arises from combining one-sided duo factor rings and centers. A ring $R$ is called {\it 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 $\pi$-duo\textbf{,} 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 $D_3(R)$ is right CIFD, where $D_3(R)$ is the ring of $3$ by $3$ upper triangular matrices over $R$ whose diagonals are equal. Furthermore\textbf{,} 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 $R$.
Abstract
This article concerns a ring property that arises from combining one-sided duo factor rings and centers. A ring $R$ is called {\it 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 $\pi$-duo\textbf{,} 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 $D_3(R)$ is right CIFD, where $D_3(R)$ is the ring of $3$ by $3$ upper triangular matrices over $R$ whose diagonals are equal. Furthermore\textbf{,} 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 $R$.
참고문헌 (18)
- A. Badawi, "On abelian π-regular rings", Comm. Algebra, 25(4), pp.1009~1021 (1997)[DOI]
- A. W. Chatters, "On right duo p. p. rings", Glasgow Math. J, 32(2), pp.221~225 (1990)[DOI]
- Hongying Chen, "Structures Related to Right Duo Factor Rings", Kyungpook Mathematical Journal, 61(1), pp.11~21 (2021), 경북대학교 자연과학대학 수학과
- 최광진, "Reversibility and symmetry over centers", 대한수학회지, 56(3), pp.723~738 (2019), 대한수학회[DOI]
- C. Faith, "Algebra. II, Grundlehren der Mathematischen Wissenschaften, No. 191" (1976), Springer-Verlag
- E. H. Feller, "Properties of primary noncommutative rings", Trans. Amer. Math. Soc, 89, pp.79~91 (1958)[DOI]
- K. R. Goodearl, "von Neumann regular rings, Monographs and Studies in Mathematics" (1979), Pitman
- I. N. Herstein, "Topics in Ring Theory" (1969), The University of Chicago Press
- Y. Hirano, "On strongly bounded rings and duo rings", Comm. Algebra, 23(6), pp.2199~2214 (1995)[DOI]
- C. Y. Hong, "One-sided duo property on nilpotents", Hacet. J. Math. Stat, 49(6), pp.1974~1987 (2020)[DOI]
- C. Huh, "p. p. rings and generalized p. p. rings", J. Pure Appl. Algebra, 167(1), pp.37~52 (2002)[DOI]
- H. Jin, "Structure of rings with com"mutative factor rings for some ideals contained in their centers", Hacet. J. Math. Stat, 50(5), pp.1280~1291 (2021)[DOI]
- 김남균, "On a generalization of right duo rings", 대한수학회보, 53(3), pp.925~942 (2016), 대한수학회
- J. Lambek, "Lectures on Rings and Modules" (1966), Blaisdell Publishing Co
- Y. Lee, "On generalizations of commutativity", Comm. Algebra, 43(4), pp.1687~1697 (2015)[DOI]
- N. H. McCoy, "Generalized regular rings", Bull. Amer. Math. Soc, 45(2), pp.175~178 (1939)[DOI]
- M. B. Rege, "Armendariz rings", Proc. Japan Acad. Ser. A Math. Sci, 73(1), pp.14~17 (1997)[DOI]
- G. Shin, "Prime ideals and sheaf representation of a pseudo symmetric ring", Trans. Amer. Math. Soc, 184, pp.43~60 (1973)[DOI]