Related papers: Higher power polyadic group rings

We introduce a broader class of nonassociative Ore extensions that unifies and generalizes several earlier constructions. We prove generalizations of Hilbert's Basis Theorem for this class, showing that they arise immediately from the…

Higher braiding gates, a new kind of quantum gate, are introduced. These are matrix solutions of the polyadic braid equations (which differ from the generalized Yang-Baxter equations). Such gates support a special kind of multi-qubit…

This paper investigates atomic factorizations in the monoid $\mathcal I(R)$ of nonzero ideals of a multivariate polynomial ring $R$, under ideal multiplication. Building on recent advances in factorization theory for unit-cancellative…

A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented in block diagonal matrix form (resulting in the Wedderburn decomposition), a general form of polyadic…

Graded rings provide a natural algebraic framework for encoding symmetry via decompositions into homogeneous components indexed by a group, together with multiplication rules reflecting the group operation. Among graded rings, strongly…

A notion of one-dimensional formal ring is presented. It consists of a triple $(A,\Phi,\Psi)$ where $A$ is a unital ring and $\Phi$ and $\Psi$ are two formal power series in $2$ variables ${\Phi(x,y),\Psi(x,y)\in A\llbracket…

Let $M$ be a multiplicative monoid with identity. Then I show that there is a universal one dimensional formal group law equipped with an action of $M$. If $M$ is $p$-perfect (i.e. $m\mapsto m^p$ is an isomorphism for some prime number $p$)…

We develop an elementary theory of partially additive rings as a foundation of ${\mathbb F}_1$-geometry. Our approach is so concrete that an analog of classical algebraic geometry is established very straightforwardly. As applications, (1)…

We present two criteria for a group $G$ to satisfy the following statements: any $G$-graded gr-prime (gr-semiprime) right gr-Goldie ring admits a gr-semisimple graded right classical quotient ring. The criterion for gr-semiprime rings is…

We present here a version of the Artin-Schreier polynomial that works in any characteristic. Let $C_p$ be the cyclic group of prime order $p$. Equivalently, we prove one can lift degree $p$ cyclic $C_p$ extensions over local rings $R,M$…

In this article, for a polyadic group(G,f),derived from group G by automorphism G and element b, we give a necessary and sufficient condition in terms of the group, the automorphism G, and the element b, in order that the polyadic group…

Algebraic structures in which the property of commutativity is substituted by the mediality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or…

The additive monoid $R_+(x)$ is defined as the set of all nonnegative integer linear combinations of binomial coefficients $\binom{x}{n}$ for $n \in \mathbb Z_+$. This paper is concerned with the inquiry into the structure of $R_+(\alpha)$…

In this paper, we consider graded near-rings over a monoid $G$ as a generalizations of graded rings over groups. We introduce certain innovative graded prime ideals and study some of its basic properties over graded near-rings.

This paper contains the results collected so far on polynomial composites in terms of many basic algebraic properties. Since it is a polynomial structure, results for monoid domains come in here and there. The second part of the paper…

The classical Riordan groups associated to a given commutative ring are groups of infinite matrices (called Riordan arrays) associated to pairs of formal power series in one variable. The Fundamental Theorem of Riordan Arrays relates matrix…

We extend a few fundamental aspects of the classical theory of non-unique factorization, as presented in Geroldinger and Halter-Koch's 2006 monograph on the subject, to a non-commutative and non-cancellative setting, in the same spirit of…

Let $R = k[x_1,\ldots, x_d]$ denote a standard graded polynomial ring over an algebraically closed field $k$, and let $I \subset R$ be a perfect ideal of codimension $2$ with an $n\times (n-1)$ linear presentation matrix $\phi$. We prove an…

Let $n$ be a positive integer and $R=(M_{ij})_{1\leq i,j\leq n}$ be a generalized matrix ring. For each $1\leq i,j\leq n$, let $I_i$ be an ideal of the ring $R_i:=M_{ii}$ and denote $I_{ij}=I_iM_{ij}+M_{ij}I_j$. We give sufficient…

We develop layered monoidal theories -- a generalisation of monoidal theories combining formal descriptions of a system at different levels of abstraction. Via their representation as string diagrams, monoidal theories provide a graphical…