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In this paper, we introduce related comparability for exchange ideals. Let be an exchange ideal of a ring . If satisfies related comparability, then for any regular matrix , there exist left invertible and right invertible such that for idempotents .
A semiring S in 𝕊𝕃⁺ is a t-k-Archimedean semiring if for all a,b ∈ S, b ∈ √(Sa) ∩ √(aS). Here we introduce the t-k-Archimedean semirings and characterize the semirings which are distributive lattice (chain) of t-k-Archimedean semirings. A semiring S is a distributive lattice of t-k-Archimedean semirings if and only if √B is a k-ideal, and S is a chain of t-k-Archimedean semirings if and only if √B is a completely prime k-ideal, for every k-bi-ideal B of S.
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