Hilbert's basis theorem for a semiring of skew polynomials

Semirings of skew polynomials are studied. Such semirings are generalizations of both polynomial semirings and skew polynomial rings. Let $\varphi$ be an endomorphism of a semiring $S$. The left semiring of skew polynomials over $S$ is the set of polynomials of the form $f=a_0+a_1x+\ldots +a_kx^k$, $a_i\in S$, with the usual addition and the multiplication given by the rule $xa=\varphi (a)x$. It is known that the semiring of polynomials over a Noetherian semiring does not have to be Noetherian. In 1976, L. Dale introduced the notion of monic ideal of a polynomial semiring $S[x]$ over a commutative semiring, i.e., of an ideal that together with any its polynomial $f=\ldots+ax^k+\ldots$ contains each monomial $ax^k$. It was shown that the Noetherian property of a semiring $S$ implies the ascending chain condition for the monic ideals from $S[x]$. We study the monic ideals of the semiring of skew polynomials $S[x,\varphi]$. To describe them, we define $\varphi$-chains of coefficient sets of ideals from the semiring $S[x,\varphi]$. The main result of the paper is the following fact: if $\varphi$ is an automorphism, then the semiring $S$ is left (right) Noetherian if and only if $S[x,\varphi]$ satisfies the ascending chain condition for the left (right) monic ideals. Examples are given showing that the injectivity of the endomorphism $\varphi$ is not sufficient for the validity of the formulated result.