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Paper   IPM / M / 8543
School of Mathematics
  Title:   Weak generators for classes of R-modules
  Author(s):  M. R. Vedadi (Joint with P. F. Smith)
  Status:   To Appear
  Journal: East-West J. Math.
  Supported by:  IPM
  Abstract:
Let R be a ring. An R-module M is called a weak generator for a class C of R-modules if HomR(M, V) is non-zero for every non-zero module V in C. A projective module M is a weak generator for C if and only if MMA for every annihilator A of a non-zero module V in C. Given any class C of R-modules, a finitely annihilated R-module M is a weak generator for the class of injective hulls of modules in C if and only if the R-module R/A is a weak generator for C, where A is the annihilator of M. Moreover a finitely annihilated R-module M is a weak generator for the class of all injective R-modules if and only if the annihilator of M is a left T-nilpotent ideal. In case the ring R is commutative, a finitely generated R-module M is a weak generator for the class of all R-modules if and only if M is a weak generator for the class of injective R-modules. In addition, if the ring R is Morita equivalent to a commutative semiprime Noetherian ring, then M is a weak generator for the class of all R-modules if and only if the trace of M in R is an essential right ideal of R.

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