Abstract
A semiring $S$ is said to be a $t$-$k$-simple semiring if it has no non-trivial proper left $k$-ideal and no non-trivial proper right $k$-ideal. We introduce the notion of $t$-$k$-simple semirings and characterize the semirings in $\mathbb{SL^{+}}$, the variety of all semirings with a semilattice additive reduct, which are distributive lattices of $t$-$k$-simple subsemirings. A semiring $S$ is a distributive lattice of $t$-$k$-simple subsemirings if and only if every $k$-bi-ideal in $S$ is completely semiprime $k$-ideal. Also the semirings for which every $k$-bi-ideal is completely prime has been characterized.
Citation
Tapas Kumar Mondal. Anjan Kumar Bhuniya. "Semirings which are distributive lattices of $t$-$k$-simple semirings." Tbilisi Math. J. 8 (2) 149 - 157, December 2015. https://doi.org/10.1515/tmj-2015-0018
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