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Ideals in the Integral Octaves Daniel Allcock*
 

Summary: Ideals in the Integral Octaves
Daniel Allcock*
12 March 1998, revised 9 May 1998
allcock@math.utah.edu
web page: http://www.math.utah.edu/łallcock
Department of Mathematics
University of Utah
Salt Lake City, UT 84112.
Abstract.
We study the nonassociative ring of integral octaves (or Cayley numbers or octonions) discovered
independently by Dickson and Coxeter. We prove that every one≠sided ideal is in fact two≠sided,
principal, and generated by a rational integer.
1 Introduction
The nonassociative ring K of integral octaves is a discrete subring of the nonassociative field O
of octaves---it is the natural analogue of the the rational integers in Z. Dickson introduced K in
[4]; one can use K to construct the finite simple groups now called G 2 (p) for p a prime number.
Much later, Coxeter [3] rediscovered the ring and obtained a number of new results concerning it.
The purpose of this note is to obtain a complete description of the ideals in K: every one≠sided
ideal is actually two≠sided and generated by a rational integer. This substantially improves a result
of Mahler [5], who proved that any one≠sided ideal is generated by a rational integer multiple of

  

Source: Allcock, Daniel - Department of Mathematics, University of Texas at Austin

 

Collections: Mathematics