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On prime Goldie-like quadratic Jordan matrix algebras

Daniel J. Britten

1977
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Canadian mathematical bulletin
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In [1] and [2] , there was given a characterization for linear Jordan matrix algebras whose coordinatizing ring is *-prime Goldie or a Cayley-Dickson ring (C-D ring). If one considers the corresponding question in the more general setting of quadratic Jordan algebra as defined by McCrimmon in [11] , then the result is similar. In this latter case the ample quadratic Jordan algebras, as studied by Montgomery in [12] and [13], are brought into play. Here we tie these concepts together in
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... ogether in extending [2, Theorem 0] to its quadratic Jordan generalization. This paper also shortens some arguments of [2] and indicates some corrections necessary in [2]. The corresponding setting is that of the Coordinatization Theorem for quadratic Jordan algebras [11] . Let K be a commutative associative ring with 1 and R an alternative K-algebra with 1 and involution *. A quadratic Jordan subalgebra T of the symmetric elements of the nucleus of R which contains the norm xx* and the trace x + x* of each element x in R is said to be ample. An ample sub-algebra T is said to be closed ample if x*TxçT for each xeR. Let R 0 be a closed ample subalgebra containing 1 and let a = diag(ai,..., a") e R n with a t invertible elements contained in R 0 . Let y a be the diagonal involution on R n given by y y° = a~1y* t a where y* r denotes the taking of conjugate transpose. Let J = H(R n , R 0 , y a ) be the set of those y a -symmetric elements' y = y 7a whose diagonal entries y it lie in a~[ 1 R 0 = Ro^i-If Ro is the set of all symmetric elements of JR which are contained in the nucleus, then we write H(R n , y a ). Ju={x[ii] = xei i :xeai 1 Ro} and J ii ={x[ij] = xei i + aJ 1 x*a i :xeR}, iV/, are the Pierce components of / where {e i; -} is a standard set of matrix units. For a subset A of J we use A tj to denote A n J iy . A quadratic (inner) ideal Q is said to be fj-quadratic if Q,-/^0. We will use the definitions of: Jordan ring of quotients; common multiple property (cmp); and the Goldie-like conditions as stated in [2]. First we point out that condition (iii) of [2, Theorem 0] should have added to it the statement that (JR, *) is a subring of (R\ *). This is used, for example, in the proof of Lemma 2. It is not necessary if n>3.

doi:10.4153/cmb-1977-008-0
fatcat:srxdzhavhjdzpg353i6jnszo74