Weakly regular modules over normal rings

Under study are some conditions for the weakly regular modules to be closed under direct sums and the rings over which all modules are weakly regular. For an arbitrary right $R$-module $M$, we prove that every module in the category $\sigma(M)$ is weakly regular if and only if each module in $\sigma(M)$ is either semisimple or contains a nonzero $M$-injective submodule. We describe the normal rings over which all modules are weakly regular.