Related papers: Additive unit representations in global fields - A…
Let $RG$ be the gruop ring of the group $G$ over ring $R$ and $\mathscr{U}(RG)$ be its unit group. Finding the structure of the unit group of a finite group ring is an old topic in ring theory. In, G. Tang et al: Unit Groups of Group…
We investigate circular planar nearrings constructed from finite fields as well the complex number field using a multiplicative subgroup of order $k$, and characterize the overlaps of the basic graphs which arise in the associated…
We introduce an approach to the categorification of rings, via the notion of distributive categories with negative objects, and use it to lay down categorical foundations for the study of super, quantum and non-commutative combinatorics.…
The aim of this project is to attach a geometric structure to the ring of integers. It is generally assumed that the spectrum $\mathrm{Spec}(\mathbb{Z})$ defined by Grothendieck serves this purpose. However, it is still not clear what…
Suppose we are given black-box access to a finite ring R, and a list of generators for an ideal I in R. We show how to find an additive basis representation for I in poly(log |R|) time. This generalizes a quantum algorithm of Arvind et al.…
We establish basic results on subrings of finite commutative rings and closely related rings. Among other applications we calculate the number of maximal subrings of a finite commutative local ring.
We enumerate the number of complex irreducible representations of each degree of general unitary groups of degree 4 over principal ideal rings of length 2.
Degree bounds for algebra generators of invariant rings are a topic of longstanding interest in invariant theory. We study the analogous question for field generators for the field of rational invariants of a representation of a finite…
Theory of representations of F-algebra is a natural development of the theory of F-algebra. Exploring of morphisms of the representation leads to the concepts of generating set and basis of representation. In the book I considered the…
We study nonmatrix varieties of $\mathbf{k}$-algebras, where $\mathbf{k}$ is a unital commutative ring. Our results extend to this generality known results for the case in which $\mathbf{k}$ is an infinite field. Also, we generalize these…
For commutative rings, we introduce the notion of a {\em universal grading}, which can be viewed as the "largest possible grading". While not every commutative ring (or order) has a universal grading, we prove that every {\em reduced order}…
We establish that all rings of $S$-integers are universally definable in function fields in one variable over certain ground fields including global and non-archimedean local fields. That is, we show that the complement of such a ring of…
A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…
What are all rings $R$ for which $R^*$ (the group of invertible elements of $R$ under multiplication) is an elementary abelian $p$-group? We answer this question for finite-dimensional commutative $k$-algebras, finite commutative rings,…
For arbitrary F-algebra, in which the operation of addition is defined, I explore biring of matrices of mappings. The sum of matrices is determined by the sum in F-algebra, and the product of matrices is determined by the product of…
We describe the "generic" part of the character ring of general linear groups over a finite field in terms of quiver representations.
Algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out consequences of this structure by means of various mathematical tools -- the theory of…
We investigate generalized quadratic forms with values in the set of rational integers over quadratic fields. We characterize the real quadratic fields which admit a positive definite binary generalized form of this type representing every…
We consider the problem of characterizing all number fields $K$ such that all algebraic integers $\alpha\in K$ can be written as the sum of distinct units of $K$. We extend a method due to Thuswaldner and Ziegler that previously did not…
This paper establishes mixed multiplicity formulas concerning the relationship between mixed multiplicities of modules and mixed multiplicities of rings via rank of modules.