A left R-module M is called weakly prime module if the annihilator of any nonzero submodule of M is a prime ideal and a proper submodule P of M is called weakly prime submodule if the quotient module M/P is a weakly prime module. This notion is introduced and extensively studied. The module in which, weakly prime submodules and the prime submodules coincide, are studied, and it is shown that multiplicative modules have this property called compatibilility property. It is also shown that each R-module is compatible if and only if each prime ideal is maximal or if and only if the R-module R@R is compatible.Over commutative rings modules in which every proper submodule(proper nonzero submodules) is
weakly prime are characterized. It is proved that if dimR < 1, then each R-module has a prime submodule if and only if it has weakly prime submodule.

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Important Note: In the following papers we have used “classical prime” and not “weakly prime” although it is “weakly prime” that many authors use, cf. Behboodi (2006), Azizi (2008) and Azizi (2009). In defense of our nomenclature, weakly prime modules (submodules) already exist in literature when used in a totally different context - the context which tries to generalize the notion of weakly prime ideals for rings to modules, cf.Ebrahimi and Farzalipour (2007). To the best of our knowledge, classical prime has not been used by other authors to mean something different.

[1] M. Behboodi, Classical prime submodules, PhD Thesis, Chamran University Ahvaz Iran (2004).

[2]-M. Behboodi, A generalization of Bear's lower nilradical for modules, J. Algebra Appl. 6(2) (2007), 337-353.

[3]- M. Behboodi (with M. Baziar), Classical primary submodules and decomposition of modules, J. Algebra Appl. 8(3) (2009), 351–362.

[4]- M. Behboodi (with M. J. Noori), Zariski-like topology on the classical prime spectrum of a module, Bull. Iranian Math. Soc.35(1) (2009), 255-271.

[5]- M. Behboodi (with S. H. Shojaee), On chains of classical prime submodules and dimension theory of modules, Bull. Iranian. Math. Soc. 36 (1) (2010), 149-166.