By Patrick J. Fleury (auth.)

ISBN-10: 3540115978

ISBN-13: 9783540115977

ISBN-10: 3540393714

ISBN-13: 9783540393719

**Read or Download Advances in Non-Commutative Ring Theory: Proceedings of the Twelfth George H. Hudson Symposium Held at Plattsburgh, USA, April 23–25, 1981 PDF**

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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**

**Sample text**

G. 1, then M suffices M --+ R n which 0 assumed exact shows as M is since R is a is torsionless. by T h e o r e m 3. 1. (Thus, of this 9enerated left remark to be finitely in mod-R, that to prove Bass' The proof generated then inasmuch 2. 1, O is left Noetherian. This from T h e o r e m M* implies is finitely as stated, result referred shows right R - m o d u l e M c--~Rn generated. ) to earlier. that the dual m o d u l e M embeds M* in a finitely free module. ACKNOWLEDGEMENT This paper was discussed impromptu during the problem session.

M correspond to the has finite length if and only if M is called o-closed if O-torsion. Recall that the o-closed submodules of Thus in the quotient category if R-Mad/a, M [14, the object satisfies both ascending and descendin~ chain conditions on the set of o-closed submodules. In this case, M/G(M) must have finite uniform dimension. If RX is an injective module, then X defines a torsion radical rad X by setting radx(M) = {m • M]f(m) = 0 for any module R-Mad/a, and M a in R-Mad. f • HomR(M,X)} o - radx, then X = 0a(X) is a cogenerator for is said to be the torsion radical' cogenerated by The torsion radical ~M If for all such that Qo(M) is said to be prime if a is a simple object in a = radE(M) R-Hod/a.

R The Walkers has finite reduced rank (on the left) and satisfies to R, then S R the re~ularitv has finite reduced rank and R. 6], and furthermore, torsion radical. S rin~ R holds for any rin~ with Krull dimension by in this case every torsion radical is contained Theorem 2 below shows that these conditions hold for fact, rings with finite reduced rank can be characterized torsion radicals there is torsion radicals of R-Mod and minimal any ring with finite reduced rank whose prime radical is riRht T-nilpotent.

### Advances in Non-Commutative Ring Theory: Proceedings of the Twelfth George H. Hudson Symposium Held at Plattsburgh, USA, April 23–25, 1981 by Patrick J. Fleury (auth.)

by George

4.5