A nicer shape of the condition $(\Lambda\Psi)$ (which ensures preservation of separation axioms $T_k$, $k=0,1,2,3,3\frac12$, in reduced ideal-products) is given. If an reduced-ideal products is $T_0$, $T_1$ or $T_2$ then ``almost all" coordinate spaces have this property. This implication holds for $T_3$-property if the condition $(\Lambda\Psi)$ is satisfied. Some results on mappings and homogenicity of r.i. products are obtained. Finally, it is proved that r.i.p. of topological groups (rings) is a topological group (ring).