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Abstract
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Modules (cogenerated by nonzero their submodules) having nonzero square homomorphism
in nonzero submodules are said (prime) weakly compressible (wc). Such modules
are semiprime (i.e. they are cogenerated by their essential submodules). For many rings
R, including commutative rings, it is proved that wc modules are isomorphic submodules
of products of prime modules, and the converse holds precisely when the class of wc
modules is enveloping for mod-R or equivalently, every semiprime module is wc. Semi-
Artinian rings R have the latter property and the converse is true when R is strongly
regular. Duo Noetherian rings over which wc modules form an enveloping class are shown
to have a finite number maximal ideals. If R is Morita equivalent to a Dedekind domain,
then the class of wc R-modules is enveloping if and only if R is simple Artinian or
J(R) = 0.
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