Weakly np- Injective Rings and Weakly C2 Rings
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A ring R is called left weakly np- injective if for each non-nilpotent element a of R, there exists a positive integer n such that any left R- homomorphism from Ran to R is right multiplication by an element of R. In this paper various properties of these rings are first developed, many extending known results such as every left or right module over a left weakly np- injective ring is divisible; R is left seft-injective if and only if R is left weakly np-injective and RR is weakly injective; R is strongly regular if and only if R is abelian left pp and left weakly np- injective. We next introduce the concepts of left weakly pp rings and left weakly C2 rings. In terms of these rings, we give some characterizations of (von Neumann) regular rings such as R is regular if and only if R is n- regular, left weakly pp and left weakly C2. Finally, the relations among left C2 rings, left weakly C2 rings and left GC2 rings are given.
Abstract
A ring R is called left weakly np- injective if for each non-nilpotent element a of R, there exists a positive integer n such that any left R- homomorphism from Ran to R is right multiplication by an element of R. In this paper various properties of these rings are first developed, many extending known results such as every left or right module over a left weakly np- injective ring is divisible; R is left seft-injective if and only if R is left weakly np-injective and RR is weakly injective; R is strongly regular if and only if R is abelian left pp and left weakly np- injective. We next introduce the concepts of left weakly pp rings and left weakly C2 rings. In terms of these rings, we give some characterizations of (von Neumann) regular rings such as R is regular if and only if R is n- regular, left weakly pp and left weakly C2. Finally, the relations among left C2 rings, left weakly C2 rings and left GC2 rings are given.
참고문헌 (18)
- R. Camps, "On semi-local rings", Israel J. Math, 81, pp.203~211 (1993)
- J. L. Chen, "On General principally injective rings", Comm. Alg, 27, pp.2097~2116 (1999)
- J. L. Chen, "On regularity of rings", Alg. Colloq, 8(3), pp.267~274 (2001)
- C. Faith, "A counter example to a conjecture of Johns", Proc. Amer. Math. Soc, 116, pp.21~26 (1992)
- C. Faith, "The structure of Johns rings", Proc. Amer. Math. Soc, 120, pp.1071~1081 (1994)
- Y. C. R. Ming, "On Quasi-Frobeniusean and Artinian rings", Publications De L; institut Math ématique, 33(47), pp.239~245 (1983)
- W. K. Nicholson, "Rings with the dual of the isomorphism theorem", J. Alg, 271, pp.391~406 (2004)
- W. K. Nicholson, "Rings with projective socle", Proc. Amer. Math. Soc, 102, pp.443~450 (1988)
- W. K. Nicholson, "Principally injective rings", J. Alg, 174, pp.77~93 (1995)
- W. K. Nicholson, "Minijective rings", J. Alg, 187, pp.548~578 (1997)
- W. K. Nicholson, "Annihilators and the CS-condition", Glasgow Math. J, 40, pp.213~222 (1998)
- W. K. Nicholson, "On finitely embedded rings", Comm. Alg, 28(11), pp.5311~5315 (2000)
- W. K. Nicholson, "Weakly continuous and C2-rings", Comm. Alg, 29(6), pp.2429~2466 (2001)
- J. L. G. Pardo, "Rings with finite essential socle", Proc. Amer. Math. Soc, 125, pp.971~977 (1997)
- S. Page, "Generalizations of principally injective rings", J. Alg, 206(3), pp.706~721 (1998)
- J. C. Wei, "Nil- injective rings", Intern. Electr. J. Alg, 3, pp.1~21 (2007)
- J. C. Wei, "NPP rings, reduced rings and SNF rings", Intern. Electr. J. Alg, 4, pp.9~26 (2008)
- Y. Q. Zhou, "Rings in which certain right ideals are direct summands of annihilators", J. Aust. Math. Soc, 73, pp.335~346 (2002)