Related papers: Polynomial functions on non-commutative rings - a …
A product of cochains in a polyhedral complex is constructed. The multiplication algorithm depends on the choice of a parameter. The parameter is a linear functional on the ambient space. Cocycles form a subring of the ring of cochains,…
This work investigates the invariance of the non-necessarily finite uniform dimension and related concepts for subextensions in skew polynomial rings \mbox{$ \mathbb{S}=R[ \mathbf{\mathrm{X}}; \mathbf{\alpha} , \mathbf{\delta} ]$} of…
We study a nonlinear analogue of additive commutators, known as \textit{polynomial commutators}, defined by $p(ab) - p(ba)$ for a polynomial $p \in F[x]$ and elements $a, b$ in an algebra $R$ over a field $F$. Originally introduced by…
Let $D$ be a domain with fraction field $K$, and let $M_n(D)$ be the ring of $n \times n$ matrices with entries in $D$. The ring of integer-valued polynomials on the matrix ring $M_n(D)$, denoted ${\rm Int}_K(M_n(D))$, consists of those…
We consider complete intersection ideals in a polynomial ring over a field of characteristic zero that are stable under the action of the symmetric group permuting the variables. We determine the possible representation types for these…
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 $R_+(\alpha)$…
Let K,S,D be a division ring, an endomorphism and a S-derivation of K, respectively. In this setting we introduce generalized noncommutative symmetric functions and obtain Vieta formula and decompositions of differential operators.…
In this paper, we introduce and study two new classes of commutative rings, namely semi transitional rings and transitional rings, which extend several classical ideas arising from rings of continuous functions and their variants. A general…
Semi-free ideal rings, or semifirs, were introduced by Paul M. Cohn to study universal localizations in the non-commutative setting. We provide new examples of semifirs consisting of analytic functions in several non-commuting variables.…
Let $R$ be a commutative ring with nonzero identity. Let $\mathcal{I}(R)$ be the set of all ideals of $R$ and let $\delta : \mathcal{I}(R)\longrightarrow \mathcal{I}(R)$ be a function. Then $\delta$ is called an expansion function of ideals…
The aim of this paper is to study the probability that the commutator of an arbitrarily chosen pair of elements, each from two different subrings of a finite non-commutative ring equals a given element of that ring. We obtain several…
Let $R$ be a commutative ring with a collection of ideals $\{ N_1, N_2, \dots, N_{k-1}\}$ satisfying certain conditions, properties of the set of invertible quadratic residues of the ring $R$ are described in terms of properties of the set…
The work proves that, for three-dimensional upper triangular groups over a field of odd characteristic with an abelian unipotent subgroup, the ring of invariants is polynomial if and only if the unipotent subgroup is generated by…
A positive integer k is a length of a polynomial if that polynomial factors into a product of k irreducible polynomials. We find the set of lengths of polynomials of the form x^n in R[x], where (R, m) is an Artinian local ring with m^2 = 0.
Let $D$ be a principal ideal domain with infinite spectrum such that for every nonzero prime ideal $M$ of $D$, the residue field $D/M$ is finite. Let $K$ be the quotient field of $D$. We investigate sets of lengths in the ring of…
We propose a version of the classical shape lemma for zero-dimensional ideals of a commutative multivariate polynomial ring to the noncommutative setting of zero-dimensional ideals in an algebra of differential operators.
Given a certain factorization property of a ring $R$, we can ask if this property extends to the polynomial ring over $R$ or vice versa. For example, it is well known that $R$ is a unique factorization domain if and only if $R[X]$ is a…
A proper ideal $P$ of a commutative ring with identity is an almost prime ideal if $ab \in P{\setminus}P^2$ implies $a \in P$ or $b \in P$. In this paper we define almost prime ideals of a noncommutative ring, and provide some equivalent…
Some Dirichlet-like functions, attached to a pair (periodic function, polynomial) are introduced and studied. These functions generalize the standard Dirichlet L-functions of Dirichlet characters. They have similar properties, being…
We develop a theory of formal multivariate polynomials over commutative rings by treating them as ring terms. Our main result is that two ring terms are s-equivalent (when expanded they yield the same standard polynomial) iff they are…