Subtractive extension of ideals in semirings
DOI:
https://doi.org/10.5644/SJM.10.1.02Keywords:
Semiring, subtractive ideal, $Q$-ideal, quotient semiring, subtractive extension of an ideal, semiprime idealAbstract
In this paper, we (1) obtain the $k$-closure of ideals and a characterization of subtractive extension of ideals in the semiring $\mathbb{Z}_0^+$; (2) introduce the concept of closure of an ideal $A$ of a semiring $R$ with respect to an ideal $I$ of $R$ and prove the set of all subtractive extensions of an ideal $I$ of a semiring $R$ is a complete lattice; (3) show that a subtractive extension $P$ of a $Q$-ideal $I$ in a semiring $R$ is a semiprime ideal if and only if $P/I_{(Q\cap P)}$ is a semiprime ideal in the quotient semiring $R/I_{(Q)}.$
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