Right Buchberger algorithm over bijective skew PBW extensions

In this paper we present a right version of the algorithms developed for to compute Gr\"obner bases over bijective skew PBW extensions in the left case given in [3]. In particular, we adapt the theory of reduction and we build a right division algorithm and generate a right version of Buchberger algorithm over bijective skew PBW extensions, finally we illustrate some examples using the SPBWE.lib library implemented in Maple (see [1], [4]). It is important to note that the development of this theory is fundamental to complete the SPBWE.lib library and to be able to develop many of the homological applications that arise as result of obtaining the right Gr\"obner bases over skew PBW extensions.Comment: [1] Fajardo, W., A computational Maple library for skew PBW extensions, Fundamenta Informaticae, 176, 2019, 159-191. [3] Fajardo, W., Gallego, C., Lezama, O., Reyes, A., Suarez, H., Vanegas, H., Skew PBW extensions, ISBN 978-3-030-53377-9, Springer Switzerland AG 2020. [4] Fajardo, W, Extended modules over skew PBW extensions, Ph.D. Thesis, Universidad Nacional de Colombia, Bogot\'a, 201

Noncommutative coding theory and algebraic sets for skew PBW extensions

The classical commutative coding theory has been recently extended to noncommutative rings of polynomial type. There are many interesting works in coding theory over single Ore extensions. In this review article we present the most relevant algebraic tools and properties of single Ore extensions used in noncommutative coding theory. The last section represents the novelty of the paper. We will discuss the algebraic sets arising in noncommutative coding theory but for skew $PBW$ extensions. These extensions conform a general class of noncommutative rings of polynomial type and cover several algebras arising in physics and noncommutative algebraic geometry, in particular, they cover the Ore extensions of endomorphism injective type and the polynomials rings over fields.Comment: arXiv admin note: text overlap with arXiv:2106.1208