On NI and quasi-NI rings
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Let $R$ be a ring. It is well-known that $R$ is {\it NI} if and only if $\sum_{i=0}^nRa_iR$ is a nil ideal of $R$ whenever a polynomial $\sum_{i=0}^na_ix^i$ is nilpotent, where $x$ is an indeterminate over $R$. We consider a condition which is similar to the preceding one: $\sum_{i=0}^nRa_iR$ contains a nonzero nil ideal of $R$ whenever $\sum_{i=0}^na_ix^i$ over $R$ is nilpotent. A ring will be said to be {\it quasi-NI} if it satisfies this condition. The structure of quasi-NI rings is observed, and various examples are given to situations which raised naturally in the process.
Abstract
Let $R$ be a ring. It is well-known that $R$ is {\it NI} if and only if $\sum_{i=0}^nRa_iR$ is a nil ideal of $R$ whenever a polynomial $\sum_{i=0}^na_ix^i$ is nilpotent, where $x$ is an indeterminate over $R$. We consider a condition which is similar to the preceding one: $\sum_{i=0}^nRa_iR$ contains a nonzero nil ideal of $R$ whenever $\sum_{i=0}^na_ix^i$ over $R$ is nilpotent. A ring will be said to be {\it quasi-NI} if it satisfies this condition. The structure of quasi-NI rings is observed, and various examples are given to situations which raised naturally in the process.
참고문헌 (13)
- G.F. Birkenmeier, "A connection between weak regularity and the simplicity of prime factor rings", Proc. Amer. Math. Soc, 122, pp.53~58 (1994)
- J.L. Dorroh, "Concerning adjunctins to algebras", Bull. Amer. Math. Soc, 38, pp.85~88 (1932)
- K. R. Goodearl, "Von Neumann Regular Rings" (1979), Pitman
- D. Handelman, "Strongly prime rings", Tran. Amer. Math. Soc, 211, pp.209~223 (1975)
- C.Y. Hong, "On minimal strongly prime ideals", Comm. Algebra, 28, pp.4867~4878 (2000)
- C. Huh, "On rings whose strongly prime ideals are com-pletely prime", Comm. Algebra, 26, pp.595~600 (1998)
- C. Huh, "On rings whose strongly prime ideals aAre com-pletely prime", Algebra Colloq, 17, pp.283~294 (2010)
- S.U. Hwang, "Structure and topological conditions of NI rings", J. Algebra, 302, pp.186~199 (2006)
- Young Cheol Jeon, "On weak Armendariz rings", 대한수학회보, 46(1), pp.135~146 (2009), 대한수학회
- N.K. Kim, "An ascending chain condition on Wedderburn radicals", Comm. Algebra, 34, pp.37~50 (2006)
- J. Lambek, "On the representation of modules by sheaves of factor modules", Canad. Math. Bull, 14, pp.359~368 (1971)
- G. Marks, "On 2-primal Ore extensions", Comm. Algebra, 29, pp.2113~2123 (2001)
- L.H. Rowen, "Ring Theory" (1991), Academic Press