A representation theorem for certain connected rings

A General Representation Theorem for Partially Ordered Commutative Rings

An extension of the Kadison-Dubois representation theorem is proved. This extends both the classical version [3] and the preordering version given by Jacobi in [5]. It is then shown how this can be used to sharpen the results on representations of strictly positive polynomials given by Jacobi and Prestel in [6]. In [4] Dubois extends a result of Kadison on representation of archimedean partiall...

Certain Congruence Criteria Connected with Fermat's Last Theorem.

Commutativity for a Certain Class of Rings

We discuss the commutativity of certain rings with unity 1 and one-sided s-unital rings under each of the following conditions: xr[xs, y] = ±[x, yt]xn, xr[xs, y] = ±xn[x, yt], xr[xs, y] = ±[x, yt]ym, and xr[xs, y] = ±ym[x, yt], where r, n, and m are non-negative integers and t > 1, s are positive integers such that either s, t are relatively prime or s[x, y] = 0 implies [x, y] = 0. Further, we ...

A Commutativity Theorem for Associative Rings

Let m > 1; s 1 be xed positive integers, and let R be a ring with unity 1 in which for every x in R there exist integers p = p(x) 0; q = q(x) 0;n = n(x) 0; r = r(x) 0 such that either x p x n ; y]x q = x r x; y m ]y s or x p x n ;y]x q = y s x; y m ]x r for all y 2 R. In the present paper it is shown that R is commutative if it satisses the property Q(m) (i.e. for all x; y 2 R;mx; y] = 0 implie...

A Dirichlet Theorem for Polynomial Rings

We prove a Dirichlet theorem for polynomial rings: Let F be a pseudo algebraically closed field (i.e., each nonempty variety defined over F has an F-rational point). Then for all relatively prime polynomials a(X), b(X) ∈ F [X] and for every sufficiently large integer n there exist infinitely many polynomials c(X) ∈ F [X] such that a(X) + b(X)c(X) is irreducible of degree n, provided that F has ...