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 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.

Tbilisi Mathematical Journal, Vol. 8(2) (2015), pp. 149-157