Coughlin, Raymond; Rich, Michael Associo-symmetric algebras. (English) Zbl 0243.17002 Trans. Am. Math. Soc. 164, 443-451 (1972). Summary: Let \(A\) be an algebra over a field \(F\) satisfying \((x,x,x) = 0\) with a function\(g: A \times A \times A \to F\) such that \((xy)z = g(x,y,z)x(yz)\) for all \(x, y, z\) in \(A\). If \(g({x_1},{x_2},{x_3}) = g(x_{1\pi },x_{2\pi},x_{3\pi })\) for all \(\pi\) in \(S_3\) and all \(x_1,x_2,x_3\) in \(A\) then \(A\) is called an associo-symmetric algebra. It is shown that a simple associo-symmetric algebra of degree \(>2\) or degree \(=1\) over a field of characteristic \(\neq 2\) is associative. In addition a finite-dimensional semisimple algebra in this class has an identity and is a direct sum of simple algebras. Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 2 Documents MSC: 17A05 Power-associative rings 17A30 Nonassociative algebras satisfying other identities Keywords:associo-symmetric; power-associative; orthogonal idempotents; semisimple; degree; principal idempotent PDFBibTeX XMLCite \textit{R. Coughlin} and \textit{M. Rich}, Trans. Am. Math. Soc. 164, 443--451 (1972; Zbl 0243.17002) Full Text: DOI References: [1] A. A. Albert, Power-associative rings, Trans. Amer. Math. Soc. 64 (1948), 552 – 593. · Zbl 0033.15402 [2] A. A. Albert, A theory of power-associative commutative algebras, Trans. Amer. Math. Soc. 69 (1950), 503 – 527. · Zbl 0039.26501 [3] Louis A. Kokoris, New results on power-associative algebras, Trans. Amer. Math. Soc. 77 (1954), 363 – 373. · Zbl 0057.27203 [4] Frank Kosier, On a class of nonflexible algebras, Trans. Amer. Math. Soc. 102 (1962), 299 – 318. · Zbl 0286.17003 [5] Robert H. Oehmke, Commutative power-associative algebras of degree one, J. Algebra 14 (1970), 326 – 332. · Zbl 0194.34302 · doi:10.1016/0021-8693(70)90108-0 [6] Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York-London, 1966. · Zbl 0145.25601 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.