By Patrick J. Fleury (auth.)
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Extra resources for Advances in Non-Commutative Ring Theory: Proceedings of the Twelfth George H. Hudson Symposium Held at Plattsburgh, USA, April 23–25, 1981
If R-Mad), T. R-Mod/a injective module is a cogenerator of By if q(M) ! 4], a is not a < T implies O is maximal if is nonzero and each nonzero a-torsionfree R-Mod/a. The following conditions are equivalent. The ring (2) There exist maximal torsion radicals R R has finite reduced rank (on the left). has finite length in for each prime ideal (i) = > ~ ~ T is the largest torsion radical such (1) Proof. Note that is a prime left Goldie ring), a is said to be maximal if then for all proper torsion radicals of In this case, any a is a nroner torsion radical (that is, and only if the quotient cateogry THEOREM (i).
Is a semiprime left be a minimal prime ideal of R, and let 43 p = radE(R/p). Then implies that R/P Furthermore, %(R) y ~ ~, so every ~-closed left ideal of is a left Goldie ring, and so must have finite length in ~ R is y-closed, which is a prime torsion radical. R-Mod/~, and so each ~-torsionfree injective module contains a submodule which maps to a simple object in the quotient category. ,Pn injective module cogenerates torsion radical. be the minimal prime ideals of radical cogenerated by R-Mod/~, E(R/Pi).
Angew. J. Proc. Amer. D. 08903. Math. Soc. , A c h a r a c t e r i z a t i o n of QF-3 rings, (1974) 533-6. Colby, Rings which have flat injectives, (1975) 239-52. , Tachikawa. 62 (1977)  Johns, Algebra  Menal, Pac. J. Matl J. Algebra 35 Injective cogenerator rings and a theorem of I,II, Proc. Amer. Math. Soc. 60 (1976) 25-30; 15-18. , A n n i h i l a t o r 5 (1967) conditions in N o e t h e r i a n rings, J. 203-221. , On the e n d o m o r p h i s m ring of a free module, preprint, Barcelona, Seccio de Matematiques, 1981.
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.)