Abstract:
In the present work two partial operations in the lattice of submodules $\boldsymbol L(_RM)$ are defined and investigated. They are the inverse operations for $\omega$-product and $\alpha$-coproduct studied in [6]. This is the continuation of the article [7], in which the similar questions for the operations of $\alpha$-product and $\omega$-coproduct are investigated.
The partial inverse operation of left quotient$N\,/_\odot\,K$ of $N$ by $K$ with respect to $\omega$-product is introduced and similarly the right quotient$N\,_:\backslash\,K$ of $K$ by $N$ with respect to $\alpha$-coproduct is defined, where $N,K\in\boldsymbol L(_RM)$. The criteria of existence of such quotients are indicated, as well as the different forms of representation, the main properties, the relations with lattice operations in $\boldsymbol L(_RM)$, the conditions of cancellation and other related questions are elucidated.
Keywords and phrases:ring, module, lattice, preradical, (co)product of preradical, left (right) quotient of submodules.
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