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Orthogonally complete alternative rings. (English) Zbl 0671.17020

A subset X of a reduced (i.e. without non-zero nilpotent elements) ring A is called orthogonal if \(ab=0\) for all \(a,b\in X\) (a\(\neq b)\), and is orthogonally complete if every orthogonal subset of A has a supremum in A where the order is defined: \(a\leq b\) iff \(ab=a^ 2\). X is called boundable if \(ab(a-b)=0\) for every \(a,b\in X\), and A is complete if every boundable subset of A has in A a supremum. An alternative ring A is said to be von Neumann regular if for any element \(a\in A\) there exists an element \(b\in A\) that \(a=aba\). The main result is the following: A von Neumann regular reduced alternative ring is complete if and only if it is orthogonally complete.
Reviewer: B.Gleichgewicht

MSC:

17D05 Alternative rings
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
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References:

[1] Abian, A., Direct product decomposition of commutative semisimple rings, (Proc. Amer. Math. Soc., 24 (1970)), 502-507 · Zbl 0207.05002
[2] Bruck, R. H.; Kleinfeld, E., The structure of alternative division rings, (Proc. Amer. Math. Soc., 2 (1951)), 878-890 · Zbl 0044.02205
[3] Burgess, W. D.; Raphael, R., Abian’s order relation and orthogonal completions for reduced rings, Pacific J. Math., 54, 55-64 (1974) · Zbl 0255.16008
[4] Burgess, W. D.; Raphael, R., Complete and orthogonally complete rings, Canad. J. Math., 27, 884-892 (1975) · Zbl 0316.13004
[5] Chacron, M., Direct product of division rings and a paper of Abian, (Proc. Amer. Math. Soc., 24 (1970)), 502-507 · Zbl 0251.16014
[6] Gonzales, S., The Complete Ring of Quotients of a Reduced Alternative Ring. Orthogonal Completion, (International Meeting on Ring Theory. International Meeting on Ring Theory, Granada, Spain. International Meeting on Ring Theory. International Meeting on Ring Theory, Granada, Spain, Comm. Alg. (1986)), to appear
[7] Gonzalez, S.; Martinez, C., Order relation in Jordan rings and a structure theorem, (Proc. Amer. Math. Soc., 98 (1986)), 379-388 · Zbl 0607.17012
[8] Haines, D. C., Injective objects in the category of \(p\)-rings, (Proc. Amer. Math. Soc., 42 (1974)), 57-60 · Zbl 0251.06027
[9] Kleinfeld, E., A characterization of the Cayley numbers, (Albert, A., Studies in Modern Algebra (1963), Math. Assoc. Amer), 126-143 · Zbl 0196.30807
[10] Myung, H. C.; Jimenez, L., Direct product decomposition of alternative rings, (Proc. Amer. Math. Soc., 47 (1975)), 53-59 · Zbl 0301.17004
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