On special \(Q_ r\)-filters of semigroups, semirings and rings
In the theory of semigroups, right quotient filters \((Q_ r\)-filters) of a semigroup S are used to construct the \(Q_ r\)-semigroup \(Q_ r(S,{\mathcal F},{\mathcal K})\) of S, depending on two of those \(Q_ r\)-filters \({\mathcal F}\) and \({\mathcal K}\). In this paper, we collect some basic properties and characterizations of special \(Q_ r\)-filters of a semigroup and define a corresponding concept for seminearrings and hence for semirings. To include rings as a subcase, we also deal with k-closed \(Q_ r\)-filters of seminearrings. It is shown that nearly all basic statements about special \(Q_ r\)- filters of semigroups are transferable to the semiring-case and, via k- closed \(Q_ r\)-filters, also to the ring-case. We use this analogy to unify the notations and to treat all three cases simultaneously. As an application of our concepts we show that the commutativity of an additively subcommutative semiring (a ring) S transfers to the semiring (ring) of right quotients \(Q_ r(S,{\mathcal F},{\mathcal K})\) of S with respect to (k-closed) \(Q_ r\)-filters \({\mathcal F}\) and \({\mathcal K}\) of S, if \({\mathcal K}\) is special. The corresponding result for semigroups is already known. Denoting by \(S^*\) the image of \(\sigma\) (S) of S under the natural homomorphism \(\sigma:S\to Q_ r(S,{\mathcal F},{\mathcal K})\) of S into a \(Q_ r\)-semigroup \((Q_ r\)-semiring, \(Q_ r\)-ring), we obtain that the image \(\sigma\) (K) of each \(K\in {\mathcal K}\) is a dense right ideal of \(S^*\), if \({\mathcal K}\) is special. From this fact we derive an embedding theorem for \(Q_ r(S,{\mathcal F},{\mathcal K})\) into a McMorris \(Q_ r\)-semigroup \((Q_ r\)-semiring, \(Q_ r\)-ring) of \(S^*\), which really contains \(S^*\) as a subalgebra.
scientific article; zbMATH DE number 3843961
scientific article