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Paper IPM / M / 8543  


Abstract:  
Let R be a ring. An Rmodule M is called a weak generator
for a class C of Rmodules if Hom_{R}(M, V) is nonzero for
every nonzero module V in C. A projective module M is a
weak generator for C if and only if M ≠ MA for every
annihilator A of a nonzero module V in C. Given any class
C of Rmodules, a finitely annihilated Rmodule M is a
weak generator for the class of injective hulls of modules in C
if and only if the Rmodule R/A is a weak generator for C,
where A is the annihilator of M. Moreover a finitely
annihilated Rmodule M is a weak generator for the class of
all injective Rmodules if and only if the annihilator of M is
a left Tnilpotent ideal. In case the ring R is commutative, a
finitely generated Rmodule M is a weak generator for the
class of all Rmodules if and only if M is a weak generator
for the class of injective Rmodules. 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
Rmodules if and only if the trace of M in R is an essential
right ideal of R.
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