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| Paper IPM / M / 8543 |
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| Abstract: | |
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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 M ≠ MA 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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