ARMENDARIZ AND QUASI-ARMENDARIZ SEMIRINGS AND PS SEMIRINGS

In this paper we extend some results of ([2], [11], [12], [13], [15]) for non commutative semirings with identity 1 ≠ 0. We prove the following theorems: (1) Let R be a CN-semiring such that 0 is a P-primary ideal of R and P2 = 0. Then R is a quasi-Armendariz semiring. (2) Let R be a semiring, M an [R, R]-bisemimodule and R′ = R ⊕ M, the trivial extension of R by M. If R is a prime semiring, then R′ is a quasi-Armendariz (resp. p.s. quasi-Armendariz) semiring if and only if M is a quasi-Armendariz (resp. p.s. quasi-Armendariz) [R, R]-bisemimodule in the sense that for f ∈ R[x] ( resp . R⟦x⟧), g ∈ M[x] ( resp . M⟦x⟧) such that fRg = 0 implies that aiRbj = 0; and gRf = 0 implies that bjRai = 0 for each coefficient ai of f and bj of g.