Related papers: Relative big polynomial rings

We consider three types of rings of supersymmetric polynomials: polynomial ones $\Lambda_{m,n}$, partially polynomial $\Lambda_{m,n}^{+y}$ and Laurent supersymmetric rings $\Lambda_{m,n}^{\pm}$. For each type of rings we give their…

A ring $R$ is said to be an almost Armendariz ring if whenever product of two polynomials in $R[x]$ is zero, then product of their coefficients are in $N_{*}(R)$. In this article, for an endomorphism $\alpha$ on $R$, we define an…

Let $\Bbbk$ be a field of characteristic $p>0$, $V$ a finite-dimensional $\Bbbk$-vector-space, and $G$ a finite $p$-group acting $\Bbbk$-linearly on $V$. Let $S = \Sym V^*$. We show that $S^G$ is a polynomial ring if and only if the…

Complexity problems associated with finite rings and finite semigroups, particularly semigroups of matrices over a field and the Rees matrix semigroups, are examined. Let M_nF be the ring of n x n matrices over the finite field F and let…

Let K be a number field and let A be its ring of integers. Let G be a connected, noncommutative, absolutely almost simple algebraic K-group. If the K-rank of G equals 2, then G(A[t]) is not finitely presented.

In the presence of a nontrivial dual Selmer group, certain global even deformation rings are shown to be finite and flat over $\mathbb{Z}_p$. Previously, flatness was only known in established cases of Langlands reciprocity in the odd…

While every matrix algebra over a field $K$ can be realized as a Leavitt path algebra, this is not the case for every graded matrix algebra over a graded field. We provide a complete description of graded matrix algebras over a field,…

Premet has conjectured that the nilpotent variety of any finite-dimensional restricted Lie algebra is an irreducible variety. In this paper, we prove this conjecture in the case of Hamiltonian Lie algebra. and show that its nilpotent…

The main goal of the paper is the discussion of a deeper interaction between matrix theory over polynomial rings over a field and typical methods of commutative algebra and related algebraic geometry. This is intended in the sense of…

Symmetric rings were introduced by Lambek to extend usual commutative ideal theory in noncommutative rings. In this paper, we study symmetric rings over which Ore extensions are symmetric. A ring R is called strongly \sigma-symmetric if the…

Computation of polynomial relative invariants is a classical tool in algebra. Relative differential invariants are central for the equivalence problem of geometric structures. We address the fundamental problem of finite generation of their…

Littlewood polynomials are polynomials with each of their coefficients in $\{-1,1\}$. A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin-Shapiro…

Given an associative, not necessarily commutative, ring R with identity, a formal matrix calculus is introduced and developed for pairs of matrices over R. This calculus subsumes the theory of homogeneous systems of linear equations with…

The coefficient algebra of a finite-dimensional Lie algebra on a finite-dimensional representation is defined as the subalgebra generated by all coefficients of the corresponding characteristic polynomial. We explore connections between…

Let $(X,L)$ be a polarized variety over a number field. We suppose that $L$ is an hermitian line bundle. Let $M$ be a non compact Riemann Surface and $U\subset M$ be a relatively compact open set. Let $\varphi:M\to X({\Bbb C})$ be a…

In this expositional paper, we discuss commutative algebra -- a study inspired by the properties of integers, rational numbers, and real numbers. In particular, we investigate rings and ideals, and their various properties. After, we…

Refining an idea of Rosenmann and Rosset we show that the now widely studied classical Leavitt algebra $L_K(1,n)$ over a field $K$ is a ring of right quotients of the unital free associative algebra of rank $n$ with respect to the perfect…

In this note, we use the concept of a polynomial ring to give an elementary proof to Cayley-Hamilton Theorem. We also give an elementary proof to Birkhoff theorem on Bi-stochastic matrices.

Possible irreducible holonomy algebras $\g\subset\osp(p,q|2m)$ of Riemannian supermanifolds under the assumption that $\g$ is a direct sum of simple Lie superalgebras of classical type and possibly of a one-dimensional center are…

In a field of Laurent series, we construct a subring which has a module structure over a Weyl algebra. Identities of Bernoulli numbers and polynomials are obtained from these algebraic structures.