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On generalized q.f.d. modules

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*M.
Saleh, S. K. Jain and S. R. Lopez-Permouth*

**Address.**

Department of Mathematics,
Birzeit University,
P.O.Box 14, West Bank, Palestine

Department of Mathematics,
Ohio University,
Athens, OH 45701, USA

**E-mail. **msaleh@birzeit.edu
jain@math.ohiou.edu
slopez@math.ohiou.edu

**Abstract.**

A right $R$-module $M$ is called a generalized {\it q.f.d.\ }module
if every M-singular quotient has finitely generated socle. In this
note we give several characterizations to this class of modules by
means of weak injectivity, tightness, and weak tightness that
generalizes the results in \cite{sanh1}, Theorem 3. In particular,
it is shown that a module $M$ is {\it
g.q.f.d}.\ iff every direct sum
of $M$-singular $M$-injective modules in ${\sigm}$ is weakly
injective iff every direct sum of $M$-singular weakly tight is
weakly tight iff every direct sum of the injective hulls of
$M$-singular simples is weakly $R$-tight.

**AMSclassification. ** 16D50, 16D60, 16D70, 16P40.

**Keywords. ** Tight, weakly tight, weakly injective, q.f.d.,
generalized q.f.d. modules, generalized weakly semisimple.