Polynomial Rings with a Pivotal Monomial

Polynomial Rings with a Pivotal Monomial1

1. Amitsur in his paper on Finite Dimensional Central Division Algebras [l] has proved that in a division ring D with center C, (P: C) 5= ra2 < =o if and only if every primitive homomorphic image of a polynomial ring P[x] is a complete matrix ring Ah, h^n, over a division ring A. Equivalently speaking, a division ring is finite dimensional over its center if and only if the polynomial ring over...

Rings with a setwise polynomial-like condition

Let $R$ be an infinite ring. Here we prove that if $0_R$ belongs to ${x_1x_2cdots x_n ;|; x_1,x_2,dots,x_nin X}$ for every infinite subset $X$ of $R$, then $R$ satisfies the polynomial identity $x^n=0$. Also we prove that if $0_R$ belongs to ${x_1x_2cdots x_n-x_{n+1} ;|; x_1,x_2,dots,x_n,x_{n+1}in X}$ for every infinite subset $X$ of $R$, then $x^n=x$ for all $xin R$.

rings with a setwise polynomial-like condition

let $r$ be an infinite ring. here we prove that if $0_r$ belongs to ${x_1x_2cdots x_n ;|; x_1,x_2,dots,x_nin x}$ for every infinite subset $x$ of $r$, then $r$ satisfies the polynomial identity $x^n=0$. also we prove that if $0_r$ belongs to ${x_1x_2cdots x_n-x_{n+1} ;|; x_1,x_2,dots,x_n,x_{n+1}in x}$ for every infinite subset $x$ of $r$, then $x^n=x$ for all $xin r$.

Differential Polynomial Rings of Triangular Matrix Rings

Differential operators on monomial rings

Rings of differential operators are notoriously difficult to compute. This paper computes the ring of differential operators on a Stanley-Reisner ring R. The D-module structure of R is determined. This yields a new proof that Nakai’s conjecture holds for Stanley-Reisner rings. An application to tight closure is described. @ 1999 Elsevier Science B.V. All rights reserved. AMS Clas.@cation: Prima...