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\(k\)-bi-ideals and quasi \(k\)-ideals in \(k\)-Clifford and left \(k\)-Clifford semirings. (English) Zbl 1289.16090

Summary: A semiring \(S\) in \(\mathbb{SL}^+\) (the variety of all semirings whose additive reduct is a semilattice) is a \(k\)-Clifford semiring (left) if and only if it is a distributive lattice of \(k\)-semifields (left). Here we characterize both the \(k\)-Clifford semirings and left \(k\)-Clifford semirings by their ideals, especially by their \(k\)-bi-ideals and quasi \(k\)-ideals. A semiring \(S\in\mathbb{SL}^+\) is a \(k\)-Clifford semiring if and only if \(I\cap B=\overline{IBI}\) for every \(k\)-ideal \(I\) and every \(k\)-bi-ideal \(B\) of \(S\).

MSC:

16Y60 Semirings
16D25 Ideals in associative algebras
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