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We investigate prime submodules of regular modules. In particular, we shall see that every proper submodule of a regular module M over an integral domain is prime. Although (von Neumann) regular rings are zero dimensional (primes ideals are maximal), it is not true for regular modules in general. We show that cyclic regular module are zero dimensional. Furthermore, it is proved that for a regular module every proper submodule is a radical submodule.