On inverse operations in the lattices of submodules

In the lattice ${\boldsymbol{L}}(_RM)$ of submodules of an arbitrary left $R$-module ${}_RM$ four operation were introduced and investigated in the paper [3]. In the present work the approximations of inverse operations for two of these operations (for $\alpha$-product and $\omega$-coproduct) are defined and studied. Some properties of left quotient with respect to $\alpha$-product and right quotient with respect to $\omega$-coproduct are shown, as well as their relations with the lattice operations in ${\boldsymbol{L}}(_RM)$ (sum and intersection of submodules). The particular case ${}_RM= {}_RR$ of the lattice ${\boldsymbol{L}}(_RR)$ of left ideals of the ring $R$ is specified.