This paper deals with the following two properties of a module \(M\) over a commutative ring \(R\) [see \textit{T. Cheatham} and \textit{E. Enochs}, Math. Jap. 26, 9-12 (1981; Zbl 0466.16013)]: \(F\) regularity (i.e., each submodule of \(M\) is pure); \(\mathbb{Z}\) regularity (i.e., for each \(m\in M\) there is a homomorphism \(\ell:M\to \mathbb{R}\) such that \(m= \ell(m)m\)). A module is called strongly \(F\) regular if all its submodules are strongly pure. The main result of the paper shows that the following properties are equivalent for a multiplication \(R\)-module \(M\): 1. \(M\) is strongly \(F\) regular; 2. \(\text{End}_R M\) is von Neumann regular; 3. \(M\) is a \(\mathbb{Z}\) regular module over \(\text{End}_R M\).