arXiv subjects

Erwin Kleinfeld

Publications and source records attributed to Erwin Kleinfeld.

4 recordsLinked to original sources

arXiv2021

A uniform characterization of the octonions and the quaternions using commutators

Let $R$ be a ring with ${\bf 1}$ which is not commutative. Assume that a non-zero commutator in $R$ is not a zero divisor. Assume further that either $R$ is alternative, but not associative, or $R$ is associative and any commutator $v\in R$ satisfies: $v^2$ is in the center of $R.$ We prove that $R$ has no zero divisors. Furthermore, if $\text{char}(R)\ne 2,$ then the localization of $R$ at its center is an octonion division algebra, if $R$ is alternative and a quaternion division algebra, if $R$ is associative. Our proof in both cases is essentially the same and it is elementary and rather self contained.

math.RA↗

arXiv2021

A characterization of the quaternions using commutators

Let $R$ be an associative ring with ${\bf 1}$ which is not commutative. Assume that any non-zero commutator $v\in R$ satisfies: $v^2$ is in the center of $R$ and $v$ is not a zero-divisor. (Note that our assumptions do not include finite dimensionality.) We prove that $R$ has no zero divisors, and that if ${\rm char(R)}\ne 2,$ then the localization of $R$ at its center is a quaternion division algebra.

math.RA↗

arXiv2021

Alternative rings whose associators are not zero-divisors

The purpose of this short note is to prove that if $R$ is an alternative ring whose associators are not zero-divisors, then $R$ has no zero divisors. By a result of Bruck and Kleinfeld, if, in addition, the characteristic of $R$ is not $2,$ then the central quotient of $R$ is an octonion division algebra over some field.

math.RA↗

arXiv2021

A short characterization of the Octonions

In this paper we prove that if $R$ is a proper alternative ring whose additive group has no $3$-torsion and whose non-zero commutators are not zero-divisors, then $R$ has no zero-divisors. It follows from a theorem of Bruck and Kleinfeld that if, in addition, the characteristic of $R$ is not $2,$ then the central quotient of $R$ is an octonion division algebra over some field. We include other characterizations of octonion division algebras and we also deal with the case where $(R,+)$ has $3$-torsion.

math.RA↗