Simple graded rings, non-associative crossed products and Cayley-Dickson doublings

TL;DR

This paper extends classical results on simple rings to non-associative graded rings, providing new insights into their structure, especially for non-associative crossed products and Cayley-Dickson doublings.

Contribution

It generalizes a key simplicity criterion to non-associative rings graded by hypercentral groups and applies it to various algebraic constructions.

Findings

Non-associative graded rings are simple iff graded simple and center is a field.

Extended Jespers' result to non-associative setting.

Provided new proofs for classical results on Cayley-Dickson doublings.

Abstract

We show that if a non-associative unital ring is graded by a hypercentral group, then the ring is simple if and only if it is graded simple and the center of the ring is a field. Thereby, we extend a result by Jespers to a non-associative setting. By applying this result to non-associative crossed products, we obtain non-associative analogues of results by Bell, Jordan and Voskoglou. We also apply our result to Cayley-Dickson doublings, thereby obtaining a new proof of a classical result by McCrimmon.