Related papers: Algebras Defined by Monic Gr\"obner Bases over Rin…

Let $K$ be a field, $\mathcal {O}_v$ a valuation ring of $K$ associated to a valuation $v$: $K\rightarrow\Gamma\cup\{\infty\}$, and ${\bf m}_v$ the unique maximal ideal of $\mathcal {O}_v$. Consider an ideal $\mathcal {I}$ of the free…

Let $K$ be an infinite field and $K< X> =K< X_1,...,X_n>$ the free associative algebra generated by $X=\{X_1,...,X_n\}$ over $K$. It is proved that if $I$ is a two-sided ideal of $K< X>$ such that the $K$-algebra $A=K< X> /I$ is almost…

Let $K$ be a field and $R=\oplus_{p\in\mathbb{N}}R_p$ an $\mathbb{N}$-graded $K$-algebra, which has an SM $K$-basis (i.e. a skew multiplicative $K$-basis) such that $R$ holds a Gr\"obner basis theory. It is proved that there is a one-to-one…

We generalize signature Gr\"obner bases, previously studied in the free algebra over a field or polynomial rings over a ring, to ideals in the mixed algebra $R[x_1,...,x_k]\langle y_1,\dots,y_n \rangle$ where $R$ is a principal ideal…

Let $R$ be an algebra over a commutative ring $k$. Suppose that $R$ is endowed with a descending filtration indexed on an ordered group $(G,<)$ such that the restriction to $k$ is positive. We show that the existence of free algebras on a…

For associative algebras in many different categories, it is possible to develop the machinery of Gr\"obner bases. A Gr\"obner basis of defining relations for an algebra of such a category provides a "monomial replacement" of this algebra.…

We explore the question concerning the number of distinct resonant algebras depending on the generator content, which consists of the Lorentz generator, translation, and new additional Lorentz-like and translation-like generators. Such…

In this paper we will define analogs of Gr\"obner bases for $R$-subalgebras and their ideals in a polynomial ring $R[x_1,\ldots,x_n]$ where $R$ is a noetherian integral domain with multiplicative identity and in which we can determine ideal…

We consider algebras over a field K defined by a presentation K <x_1,..., x_n : R >, where $R$ consists of n choose 2 square-free relations of the form x_i x_j = x_k x_l with every monomial x_i x_j, i different from j, appearing in one of…

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and $R=\mathbb{K}[x_1,x_2,...x_n]$ the polynomial ring in $n$ variables over $\mathbb K.$ We study bases of the free $R$-module $W_n(\mathbb{K})$ of all…

In this paper we introduce an algebra embedding $\iota:K< X >\to S$ from the free associative algebra $K< X >$ generated by a finite or countable set $X$ into the skew monoid ring $S = P * \Sigma$ defined by the commutative polynomial ring…

Given an algebra A, presented by generators and relations, i.e. as a quotient of a tensor algebra by an ideal, we construct a free algebra resolution of A, i.e. a differential graded algebra which is quasi-isomorphic to A and which is…

Standard noncommutative Gr\"obner basis procedures are used for computing ideals of free noncommutative polynomial rings over fields. This paper describes Gr\"obner basis procedures for one-sided ideals in finitely presented noncommutative…

We provide a self-contained introduction to Gr\"obner bases of submodules of $R[x_1, \ldots, x_n]^k$, where $R$ is a Euclidean domain, and explain how to use these bases to solve linear systems over $R[x_1, \ldots, x_n]$.

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{\sigma (1)} a_{\sigma (2)} ... a_{\sigma (n)}$, where $\sigma$…

The main objective of this paper is to connect the theory of $Gr\"obner$ bases to concepts of homological algebra. $Gr\"obner$ bases, an important tool in algebraic system and in linear algebra help us to understand the structure of an…

Let $K\left\langle X \right\rangle$ denote the free associative algebra generated by a set $X = \{x_1, \dots, x_n\}$ over a field $K$ of characteristic $0$. Let $I_p$, for $p \geq 2$, denote the two-sided ideal in $K\left\langle X…

Let $R=\oplus_{\Gamma\in\Gamma}R_{\gamma}$ be a $\Gamma$-graded $K$-algebra over a field $K$, where $\Gamma$ is a totally ordered semigroup, and let $I$ be an ideal of $R$. Considering the $\Gamma$-grading filtration $FR$ of $R$ and the…

Let $K$ be a field, let $\sigma$ be an automorphism of $K$, and let $\delta$ be a derivation of $K$. We show that if $D$ is one of $K(x;\sigma)$ or $K(x;\delta)$, then $D$ either contains a free algebra over its center on two generators, or…

Let $K<X> =K<X_1,...,X_n>$ be the free $K$-algebra on $X={X_1,...,X_n}$ over a field $K$, which is equipped with a weight $\mathbb{N}$-gradation (i.e., each $X_i$ is assigned a positive degree), and let ${\cal G}$ be a finite homogeneous…