On the semiprimitivity of free skew extensions of rings

TL;DR

This paper proves that free skew extensions of semiprime Goldie rings are semiprimitive, expanding understanding of ring extensions' properties in noncommutative algebra.

Contribution

It establishes the semiprimitivity of free skew extensions of semiprime Goldie rings, a significant result in the theory of noncommutative ring extensions.

Findings

Free skew extensions of semiprime Goldie rings are semiprimitive.

The result applies to rings with noncommuting variables and automorphisms.

The paper advances the theory of ring extensions in noncommutative algebra.

Abstract

Let $X$ be a set of noncommuting variables of cardinality $card(X)\geqslant 2$, and ${\mathscr G}=\{\sigma_x\}_{x\in X}$, ${\mathscr D}=\{\delta_x\}_{x\in X}$ be families of automorphisms and skew derivations of the ring $R$. It is proved that if the ring $R$ is semiprime Goldie, then the free skew extension $R[X;{\mathscr G},{\mathscr D}]$ is semiprimitive.