It is well known that, when \(R\) is a commutative ring with identity, the radical of a primary ideal of \(R\) is a prime ideal of \(R\). The purpose of the present paper is to made an attempt for generalizing this property to modules.
The main results can be summarized as follows: let \(R\) be a ring, if one of the following conditions holds then, for every primary submodule \(Q\) of an \(R\)-module \(M\), rad\((Q) = M\) or rad\((Q)\) is a prime submodule of \(M\):
(1) \(R\) is a ZPI-ring, or an almost multiplication ring, or an arithmetical ring satisfying locally ACC on principal ideals;
(2) \(M\) is a special module, or a secondary representable module, or a module with DCC on cyclic submodules, or a module with DCC on the submodules of the form \(\{ r^n M\mid n \in \mathbb N \}\), for each \(r \in R\).