Geodesic
On inverse operations in the lattices of submodules
Algebra and discrete mathematics, Tome 13 (2012) no. 2, pp. 273-288.

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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.
Keywords: ring, preradical, lattice, $\alpha$-product of submodules, left (right) quotient.
Mots-clés : module 
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A. I. Kashu. On inverse operations in the lattices  of submodules. Algebra and discrete mathematics, Tome 13 (2012) no. 2, pp. 273-288. http://geodesic.mathdoc.fr/item/ADM_2012_13_2_a7/

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