Tight and cotight modules
by
Mohammad Saleh
Birzeit University

Given a right R-module M, a module Q in \sigma[M] is said to be weakly injective (resp., weakly tight) in \sigma[M] if for every finitely generated submodule N of the M-injective hull [^Q], N is contained in a submodule Y of [^Q] such that Y =~ Q (resp., N is finitely Q-cogenerated). Weakly projective modules in \sigma[M] are defined dually. For a locally q.f.d. module M, there exists a module K in \sigma[M] such that K\oplusN is weakly injective in \sigma[M], for any N in \sigma[M]. Similarly, if M is projective and perfect in \sigma[M], then there exists a module K in \sigma[M] such that K\oplusN is weakly projective in \sigma[M], for any N in \sigma[M]. Several characterizations of semisimple modules are obtained using weak injectivity and weak projectivity in \sigma[M]. For some classes M of modules in \sigma[M] we study when direct sums of modules from M are (weakly) tight in \sigma[M]. In particular, we get necessary and sufficient conditions for \sum-tightness or \sum-weak tightness of the injective hull of a simple module. As a consequence, we get characterizations of q.f.d. rings by means of weakly injective (tight) modules given by Jain and López-Permouth. Also we investigate over which rings is every tight module is cotight and conversely.

Date received: January 28, 1999