By Patrick J. Fleury (auth.)

ISBN-10: 3540115978

ISBN-13: 9783540115977

ISBN-10: 3540393714

ISBN-13: 9783540393719

**Read Online 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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**Additional 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**

A module M/M' is is called a-torsionfree M' ! 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.

We say R of FBN rings are those a polynomial left stability condition if R/P If R is both right and and both right and left fully bounded, rings satisfying exists an n ( ~ ideal P. A left identity (PI rings). is equivalent ideal P satisfies to the the left Artin-Rees if I is a left ideal of the ring R, then there such that pn n I ! PI [i0, Proposition, p. 310]. 48 We say P satisfies the Artin-Rees left and right Artin-Rees R-module property property. R M, denoted by K-dim(RM), by Rentschler and Gabriel.

The proof will make use of the characterization rank given in Theorem i. has only finitely many If the torsion radical ~i is cogenerated n has finite length with respect to ~ = ni=l~ i. ,S n. RR rank (on the left), then so does Proof. R R-Mod is maximal. ,n Then if every prime torsion radical is maximal, each simple module defines a maximal nonisomorphic The then every prime torsion radical is maximal by Conversely, only finitely many maximal by is a left Noethrian ring, then be a ring with finite reduced rank (on the left) such that only if every prime torsion radical of If R this result to certain rings with finite reduced rank.

### 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.)

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