On the Semiring of Skew Polynomials over a Bezout Semiring

In the paper, we study the semiring of skew polynomials over a Rickart Bezout semiring. Namely, let every left annihilator ideal of a semiring $S$ be an ideal. Then the semiring of skew polynomials $R=S[x,\varphi]$ is a semiring without nilpotent elements, and every its finitely generated left monic ideal is principal if and only if $S$ is a left Rickart left Bezout semiring, $\varphi$ is a rigid endomorphism, and $\varphi(d)$ is invertible for any nonzerodivisor $d$. We also obtain a characterization of the semiring $R$ in terms of Pierce stalks of the semiring $S$. The structure of left monic ideals of the semiring of skew polynomials over a left Rickart left Bezout semiring is clarified.