Related papers: On the Dimension Theory of Polynomial Rings over P…
This article introduces the notion of an NJ-reflexive ring and demonstrates that it is distinct from the concept of a reflexive ring. The class of NJ-reflexive rings contains the class of semicommutative rings, the class of left (right)…
Subspace codes and particularly constant dimension codes have attracted much attention in recent years due to their applications in random network coding. As a particular subclass of subspace codes, cyclic subspace codes have additional…
We develop almost ring theory, which is a domain of mathematics somewhere halfway between ring theory and category theory (whence the difficulty of finding appropriate MSC-class numbers). We apply this theory to valuation theory and to…
UJ-rings are studied, i.e. ring in which all units can be presented in a form 1 + x, for some x\in J(R). The behavior of UJ-rings under various algebraic construction is investigated. In particular, it is shown that the problem of lifting…
We announce numerous new results in the theory of orthogonal polynomials on the unit circle.
The notion of trivial extension of a ring by a module has been extensively studied and used in ring theory as well as in various other areas of research like cohomology theory, representation theory, category theory and homological algebra.…
A three-dimensional polynomial algebra of order $m$ is defined by the commutation relations $[P_0, P_\pm]$ $=$ $\pm P_\pm$, $[P_+, P_-]$ $=$ $\phi^{(m)}(P_0)$ where $\phi^{(m)}(P_0)$ is an $m$-th order polynomial in $P_0$ with the…
Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the…
We translate notions and results of decomposition and dimension theories for module categories, into the lattice environment. In particular we translate dimension theory in module categories to complete modular upper-continuous lattices.
In this paper we get two new classes of regular sequences in the polynomial ring over the field of complex numbers.
Let $L$ be the language of rings. We provide an axiomatization of the $L$-theories of quaternions and octonions and characterize their models: they coincide, up to isomorphism, with quaternion and octonion algebras over a real closed field,…
We consider the problem of bounding away from 0 the minimum value m taken by a polynomial P of Z[X_1,...,X_k] over the standard simplex, assuming that m>0. Recent algorithmic developments in real algebraic geometry enable us to obtain a…
Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory's connections with relative homological algebra and with…
In this paper, we list several interesting structures of cyclotomic polynomials: specifically relations among blocks obtained by suitable partition of cyclotomic polynomials. We present explicit and self-contained proof for all of them,…
We improve certain degree bounds for Grobner bases of polynomial ideals in generic position. We work exclusively in deterministically verifiable and achievable generic positions of a combinatorial nature, namely either strongly stable…
For any ring we propose the construction of a cover which increases the finitistic dimension on one side and decreases the finitistic dimension to zero on the opposite side. This complements recent work of Cummings.
Let $R$ be a ring and $\sigma$ an endomorphism of $R$. In this note, we study skew polynomial rings and skew power series rings over idempotent reflexive rings and abelian rings. Also, we introduce the concept of right (resp., left)…
We extend the existing skew polynomial representations of matrix algebras which are direct sum of matrix spaces over division rings. In this representation, the sum-rank distance between two tuples of matrices is captured by a weight…
Additional remarks and questions for transseries. In particular: properties of composition for transseries; the recursive nature of the construction of R[[[ x ]]]; modes of convergence for transseries. There are, at this stage, questions…
Recent progress concerning regularization of supersymmetric theories is reviewed. Dimensional reduction is reformulated in a mathematically consistent way, and an elegant and general method is presented that allows to study the…