Insertion Property by Essential Ideals
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
We discuss the condition that if $ab=0$ for elements $a, b$ in a ring $R$ then $aIb=0$ for some essential ideal $I$ of $R$. A ring with such condition is called {\it IEIP}. We prove that a ring $R$ is IEIP if and only if $D_n(R)$ is IEIP for every $n\geq 2$, where $D_n(R)$ is the ring of $n$ by $n$ upper triangular matrices over $R$ whose diagonals are equal. We construct an IEIP ring that is not Abelian and show that a well-known Abelian ring is not IEIP, noting that rings with the insertion-of-factors-property are Abelian.
Abstract
We discuss the condition that if $ab=0$ for elements $a, b$ in a ring $R$ then $aIb=0$ for some essential ideal $I$ of $R$. A ring with such condition is called {\it IEIP}. We prove that a ring $R$ is IEIP if and only if $D_n(R)$ is IEIP for every $n\geq 2$, where $D_n(R)$ is the ring of $n$ by $n$ upper triangular matrices over $R$ whose diagonals are equal. We construct an IEIP ring that is not Abelian and show that a well-known Abelian ring is not IEIP, noting that rings with the insertion-of-factors-property are Abelian.
참고문헌 (8)
- R. Antoine, "Nilpotent elements and Armendariz rings", J. Algebra, 319, pp.3128~3140 (2008)
- H. E. Bell, "Near-rings in which each element is a power of itself", Bull. Austral. Math. Soc, 2, pp.363~368 (1970)
- J. L. Dorroh, "Concerning adjunctions to algebras", Bull. Amer. Math. Soc, 38, pp.85~88 (1932)
- K. R. Goodearl, "An Introduction to Noncommutative Noetherian Rings" (1989), Cambridge University Press
- Y. Hirano, "On rings whose prime radical contains all nilpotent elements of index two", Arch. Math, 66, pp.360~365 (1996)
- C. Huh, "Armendariz rings and semicommutative rings", Comm. Algebra, 30, pp.751~761 (2002)
- 이양, "Structure of unit-IFP rings", 대한수학회지, 55(5), pp.1257~1268 (2018), 대한수학회
- N. K. Kim, "Extensions of reversible rings", J. Pure Appl. Algebra, 185, pp.207~223 (2003)