Subtractive extension of ideals in semirings - Chaudhari, J. N.; Davvaz, B.; Ingale, K. J. (Sarajevo Journal of Mathematics, 2024)
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 $R/I_{(Q)}.$