Advances in Non-Commutative Ring Theory: Proceedings of the by Patrick J. Fleury (auth.)

By Patrick J. Fleury (auth.)

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Extra info for Advances in Non-Commutative Ring Theory: Proceedings of the Twelfth George H. Hudson Symposium Held at Plattsburgh, USA, April 23–25, 1981

Example text

Noetherian. (i). , is not right FGF. 4 R is semilocal denotes Theorem. right FGF ring Q if R/tad R is semisimple Artinian, the Jacobson radical of Any semilocal right is QF. R. (or left) Noetherian 31 Proof. since Let jn J Q. denote the Jacobson radical of Now, is an ideal, lj _C_I(O2)C~_... C_I(jn)C_ is an ascending chain of implies that ±(jn) (right) = l(jn+l) ... ideals so Q right N o e t h e r i a n for some integer right ideal is a right FGF ring n. Since any is a right annulet, we have that jn = jn+l which in a right N o e t h e r i a n ring Nakayama's lemma).

5]. left stability is not enough to insure perfect prime every ring has Krull a prime ideal P if C(P) = (r E R I r + P is regular Another of of R. 4] has proved that a noetherian (left and right) case in which R has a perfect R ring at ring Smith stable ring R has at every prime ideal P. localization at every ideal is given by the following proposition. Proposition i: A right noetherian, R can be left localized left stable FLBN ring at every prime ideal P. Moreover Rp is a left stable FLBN ring.

27(1975), 115-120. [5] , Some aspects of noncommutative localization, Noncommutative Ring Theory, Kent State, 1975, Lecture Notes in Mathematics #545, pp. 2-31. [6] , Injective modules with both ascending and descending chain conditions on annihilators, Commun. Algebra 6(1978), 1777-1788. [7] , Rings with finite reduced rank, Commun. Algebra (to appear). W. Chatters, A. W. Goldie, C. R. Hajarnavis and T. H. Lenagan, Reduced rank in Noetherian rings, J. Algebra 61(1979), 582-589. W. Goldie, Torsion-free modules and rings, J.

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