Von-Neumann Finiteness and Reversibility in some Classes of Non-Associative Algebras
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Von-Neumann Finiteness and Reversibility in some Classes of Non-Associative Algebras Erik Darpo¨ 1 · Patrik Nystedt2 Received: 8 May 2018 / Accepted: 11 August 2020 / © The Author(s) 2020
Abstract We investigate criteria for von-Neumann finiteness and reversibility in some classes of non-associative algebras. Types of algebras that are studied include alternative, flexible, quadratic and involutive algebras, as well as algebras obtained by the Cayley–Dickson doubling process. Our results include precise criteria for von-Neumann finiteness and reversibility of involutive algebras in terms of isomorphism types of their 3-dimensional subalgebras. Keywords Von-Neumann finite · Dedekind finite · Directly finite · Reversible · Quadratic algebra · Involutive algebra Mathematics Subject Classification (2010) 17D05 · 16K20 · 17C55 · 16W10 · 16U99
1 Introduction A unital ring A is called von-Neumann finite (or Dedekind finite, or weakly 1-finite, or affine finite, or directly finite, or inverse symmetric) if every one-sided inverse in A also is two-sided, in other words, if for all a, b ∈ A satisfying ab = 1, the relation ba = 1 also holds. Many different classes of associative rings have been shown to be vonNeumann finite, for instance noetherian, self-injective and PI-rings [13]. Also group rings over fields of characteristic zero [10, 18, 20], or, more generally, endomorphism rings of permutation modules over such rings [15, 16], have been shown to be von-Neumann finite. Von-Neumann finiteness for group rings of positive characteristic has, however, remained Presented by: Steffen Koenig Patrik Nystedt
[email protected] Erik Darp¨o [email protected] 1
Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8602, Japan
2
Department of Engineering Science, University West, SE-46186 Trollh¨attan, Sweden
E. Darp¨o, P. Nystedt
an open problem (for partial results, see [3] and [9]). Of course, not all associative unital rings are von-Neumann finite; for instance if V is a vector space, then it is easy to see that End(V ) is von-Neumann finite if and only if V is finite dimensional. Following Cohn [4], we say that A is reversible if for all a, b ∈ A satisfying ab = 0, the relation ba = 0 also holds. It is easy to see that the class of associative reversible rings is properly contained in the class of associative von-Neumann finite rings. Indeed, suppose that A is an associative reversible ring and ab = 1 for some a, b ∈ A. Then, since (ba − 1)b = 0, we get that b(ba − 1) = 0 that is b2 a = b which implies that ba = abba = ab = 1. Moreover, if V is a vector space satisfying 1 < dim(V ) < ∞, then End(V ) is von-Neumann finite but not reversible. In this article, we consider von-Neumann finiteness and reversibility for some classes of unital rings which are not associative – apparently a new line of investigation. Our motivation for doing this is two-fold. First, we wish to see to which extent patterns and phenomena from the associative context reproduce in more general classes of al
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