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Rings with (a; b; c) ? (a; c; b) = and (a; [b; c]; d) = 0:
A Case Study Using Albert

Irvin Roy Hentzel
Department of Mathematics
Iowa State University
Ames, Iowa 50011 USA
[email protected]

D.P. Jacobs ?
Department of Computer Science
Clemson University
Clemson, S.C. 29634-1906 USA
[email protected]
Erwin Kleinfeld
Mathematics Department
University of Iowa
Iowa City, IA 52242 USA
[email protected]
December 18, 1992

Albert is an interactive computer system for building nonassociative algebras [2]. In this paper, we suggest certain techniques for using Albert that allow one to posit and test hypotheses effectively. This process provides a fast way to achieve new results, and interacts nicely with traditional methods. We demonstrate the methodology by proving that any semiprime ring, having characteristic <> 2; 3, and satisfying the identities (a; b; c) ? (a; c; b) = (a; [b; c]; d) = 0, is associative. This generalizes a recent result by Y. Paul [7].
Key words: identity, nonassociative polynomial, nonassociative ring, algebra.
AMS (MOS) subject classifications: 17D99, 68N99
CR Categories: I.1.3 (Special-purpose algebraic systems), I.1.4 (Applications)

1 Introduction

Recently, an interactive computer program known as Albert, for building nonassociative algebras was developed [2]. With this system, the user specifies the generators and the identities that the algebra is to satisfy, as well as the underlying field of scalars. Albert constructs the free nonassociative algebra satisfying these identities. Then one may query ?This research was partially supported by NSF Grant #CCR8905534.