### On inverse operations in the lattices of submodules

#### Abstract

In the lattice \(\mathbf{L}(_{R}M)\) of submodules of an arbitrary left \(R\)-module \(_R M\) 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 \(\mathbf{L}(_{R}M)\) (sum and intersection of submodules). The particular case \(_{R}M=_{R}R\) of the lattice \(\mathbf{L}(_{R}R)\) of left ideals of the ring \(R\) is specified.

#### Keywords

ring, module, preradical, lattice, product of submodules, left (right) quotient

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