On densely imbedded ideals of algebras

The paper arose as a result of solution of a problem of finding necessary and sufficient conditions characterizing a pair of associative rings or algebras $A$, $B$, where $A$ is a densely imbedded ideal in $B$. It turns out that the methods by which one succeeds in obtaining this solution permit treatment of the even more general situation of the so-called distributive $\Omega$-semigroups, for which the corresponding $\Omega$-algebras are commutative. This situation, with $\Omega$ empty, includes the case of semigroups also.
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