M.V. Babenko, V.V. Chermnykh. Hilbert’s basis theorem for a semiring of skew polynomials ... P. 56-65

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.

Keywords: semiring of skew polynomials, monic ideal, $\varphi$-chain of coefficient sets, Hilbert's basis theorem

Received March 20, 2022

Revised March 30, 2022

Accepted April 4, 2022

Marina Vladimirovna Babenko, Department of Applied Mathematics and Computer Science, Vyatka State University, Kirov, 610000, Russia, usr11391@vyatsu.ru

Vasiliy Vladimirovich Chermnykh, Dr. Phys.-Math. Sci., Pitirim Sorokin Syktyvkar State University, Syktyvkar, 167001 Russia, vv146@mail.ru

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Cite this article as: M.V. Babenko, V.V. Chermnykh. Hilbert’s basis theorem for a semiring of skew polynomials. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 2, pp. 56–65.