#### Title

Conditions that Guarantee that all Nilpotents Commute with Every Element of an Alternative Ring

#### Document Type

Article

#### Department

Mathematics (HMC)

#### Publication Date

12-1977

#### Abstract

Below I examine the meaning of condition (*) in any (not necessarily associative) ring and show that under rather mild restrictions (e.g.,*2 ^{a} 2 = 0* and

*a ε N*imply

*a*) (*) is equivalent to (**) either every element of

^{2}= 0*N*commutes with every element of

*R*, or if this latter does not hold,

*a*for every

^{2}= 0*a ε N*, and every element of

*N*anti-commutes with every element of

*R*. In fact, (**) holds if and only if

*R*

*R*satisfies (*). A left identity element in a ring

*R*that satisfies (*) is unique, and if

*R*is alternative, then (*) implies that

*N*is always an ideal of

*R*. A variety of additional assumptions on

*R*guarantee that (*) implies that every element of

*N*commutes with every element of

*R*, but there is an associative ring with identity that satisfies (*) in which some element of

*N*fails either to commute or anti-commute with some element of

*R*.

Then, I apply these results to obtain a commutativity theorem which avoids chain conditions on *R/N* (and the redundant assumption that *N* is an ideal of *R*). Also, some results on anti-commutative rings that satisfy (*) are obtained.

#### Rights Information

© 1977 Birkhauser Verlag

#### Terms of Use & License Information

#### DOI

10.1007/BF02485421

#### Recommended Citation

Henriksen, Melvin. 1977. Conditions that guarantee that all nilpotents commute with every element of an alternative ring. Algebra Universalis. 7(1):119-132. .