Macdonald’s theorem for quadratic Jordan algebras

Lewand, Robert; McCrimmon, Kevin

doi:10.2140/pjm.1970.35.681

Cited by 20 publications

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“…Certainly z¡9t1/2 = 9l1/2z¡ = 0 implies z¡ ° 9t1/2 = 0, and the converse follows from (14), so the two definitions of 3¡ agree.…”

Section: Peirce Decomposition Theorem If E Is An Idempotent In a Nonmentioning

confidence: 95%

“…By (14) we have xii9t,y = 9tyxij = 0. For k^ij, we have x" ° %k = x" ° (91 w ° %k) (by (I')) = (Xü o Sly) o St/fc (by (16)) =0, so by (14) again xu%k = %kxu = 0. By Peirce orthogonality these are enough to show that x¡¡ annihilates St.…”

Section: Kevin Mccrimmonmentioning

confidence: 98%

“…On 9í1/2, [EXl, Fyo]=[VXiEe, VVoGe] (by (14), for G the opposite of F) =[VXl, Vyo]EeGe(by (13)) =0 since V^0 and V%1 commute in any quadratic Jordan algebra.…”

Section: Peirce Decomposition Theorem If E Is An Idempotent In a Nonmentioning

confidence: 99%

“…Here we use interconnectivity: St,, is spanned by the Pü(x2) for /#_/, and we set 7W4)) = PÁie^í)2}. We have actually shown T(xu o zik) = T(xu) o zik for any x"=Pii(2 Xi2) in 31,,; from this we obtain (ii)" r(x"jlJ)=r(x")jü since T(xii)yij = ei{T(xií) ° yíj} = eí{T(xii o y^^Tie^Xa ° ya)) (by (ii)', or since Let commutes with T=Le.Re) =T(xiiyij) using (14). Since T is linear and preserves Peirce spaces, the only other formulas necessary in addition to (i)-(n) to show T belongs to the centroid T(xy) = T(x)y = xT(y) are (iii) Tixtfyt,) = T(xv)yti = xtí7¡^, (iv) r(x" v") = T(xit)yu = xHT(yu).…”

Section: Kevin Mccrimmonmentioning

confidence: 99%

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Noncommutative Jordan Rings

McCrimmon¹

1971

Transactions of the American Mathematical Society

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Abstract. Heretofore most investigations of noncommutative Jordan algebras have been restricted to algebras over fields of characteristic ^2 in order to make use of the passage from a noncommutative Jordan algebra 91 to the commutative Jordan algebra 91+ with multiplication x-y = i(xy+yx).We have recently shown that from an arbitrary noncommutative Jordan algebra 91 one can construct a quadratic Jordan algebra 91+ with a multiplication Uxy = x(xy+yx) -x2y = (xy+yx)x-yx2, and that these quadratic Jordan algebras have a theory analogous to that of commutative Jordan algebras.In this paper we make use of this passage from 91 to 91+ to derive a general structure theory for noncommutative Jordan rings. We define a Jacobson radical and show it coincides with the nil radical for rings with descending chain condition on inner ideals; semisimple rings with d.c.c. are shown to be direct sums of simple rings, and the simple rings to be essentially the familiar ones.In addition we obtain results, which seem to be new even in characteristic ^ 2, concerning algebras without finiteness conditions. We show that an arbitrary simple noncommutative Jordan ring containing two nonzero idempotents whose sum is not 1 is either commutative or quasiassociative.1. Axioms and basic results. Throughout this paper 91 will denote a noncommutative Jordan algebra over a unital, commutative, associative ring of scalars i>; we obtain Jordan rings by taking will be completely arbitrary, and we impose no finiteness conditions on 9t.We recall the definition [9, p. 472], [11, p. 141]. A noncommutative Jordan algebra 91 is a space together with a bilinear product xy such that the flexible law hold strictly, i.e. they remain valid in any scalar extension 9tn = 9I <8>« ß. This will automatically be the case if these identities hold in 9Í and is a field with at least three elements (as in most previous investigations), and in general it merely amounts to assuming in addition the linearized version

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“…Certainly z¡9t1/2 = 9l1/2z¡ = 0 implies z¡ ° 9t1/2 = 0, and the converse follows from (14), so the two definitions of 3¡ agree.…”

Section: Peirce Decomposition Theorem If E Is An Idempotent In a Nonmentioning
confidence: 95%

Section: Kevin Mccrimmonmentioning
confidence: 98%

“…On 9í1/2, [EXl, Fyo]=[VXiEe, VVoGe] (by (14), for G the opposite of F) =[VXl, Vyo]EeGe(by (13)) =0 since V^0 and V%1 commute in any quadratic Jordan algebra.…”

Section: Peirce Decomposition Theorem If E Is An Idempotent In a Nonmentioning
confidence: 99%

Section: Kevin Mccrimmonmentioning
confidence: 99%

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Noncommutative Jordan Rings
McCrimmon¹
1971
Transactions of the American Mathematical Society
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Abstract. Heretofore most investigations of noncommutative Jordan algebras have been restricted to algebras over fields of characteristic ^2 in order to make use of the passage from a noncommutative Jordan algebra 91 to the commutative Jordan algebra 91+ with multiplication x-y = i(xy+yx).We have recently shown that from an arbitrary noncommutative Jordan algebra 91 one can construct a quadratic Jordan algebra 91+ with a multiplication Uxy = x(xy+yx) -x2y = (xy+yx)x-yx2, and that these quadratic Jordan algebras have a theory analogous to that of commutative Jordan algebras.In this paper we make use of this passage from 91 to 91+ to derive a general structure theory for noncommutative Jordan rings. We define a Jacobson radical and show it coincides with the nil radical for rings with descending chain condition on inner ideals; semisimple rings with d.c.c. are shown to be direct sums of simple rings, and the simple rings to be essentially the familiar ones.In addition we obtain results, which seem to be new even in characteristic ^ 2, concerning algebras without finiteness conditions. We show that an arbitrary simple noncommutative Jordan ring containing two nonzero idempotents whose sum is not 1 is either commutative or quasiassociative.1. Axioms and basic results. Throughout this paper 91 will denote a noncommutative Jordan algebra over a unital, commutative, associative ring of scalars i>; we obtain Jordan rings by taking will be completely arbitrary, and we impose no finiteness conditions on 9t.We recall the definition [9, p. 472], [11, p. 141]. A noncommutative Jordan algebra 91 is a space together with a bilinear product xy such that the flexible law hold strictly, i.e. they remain valid in any scalar extension 9tn = 9I <8>« ß. This will automatically be the case if these identities hold in 9Í and is a field with at least three elements (as in most previous investigations), and in general it merely amounts to assuming in addition the linearized version
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“…it will be convenient to express UbUbV"a' as ROBERT LEWAND [September an identity which is easily verified in the case of special Jordan algebras and is therefore true for all Jordan algebras as a consequence of Macdonald's theorem [4]. Now, since y(b') = (p(b') = <p(Ubb') for b' e SB, y(a o a') = <p(Ub(a ° a')) = <p(UbUh(a ° a'))…”

Section: Preliminariesmentioning
confidence: 99%

Extending a Jordan Ring Homomorphism
Lewand¹
1973
Proceedings of the American Mathematical Society
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Abstract.In this paper a homomorphism from an ideal S of a quadratic Jordan algebra 3 without 2-torsion over a ring onto a unital quadratic Jordan algebra 3' without 2-torsion is extended to a homomorphism from 3 to 3'-We then show if D is any class of quadratic Jordan algebras without 2-torsion, then the upper radical property determined by D is hereditary. Preliminaries.We adopt the notation and terminology of an earlier paper

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Local and Subquotient Inheritance of Simplicity in Jordan Systems
Anquela
¹
,
Cortés
²

2001
Journal of Algebra
14
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In this paper we prove that the local algebras of a simple Jordan pair are simple. Jordan pairs all of which local algebras are simple are also studied, showing that they have a nonzero simple heart, which is described in terms of powers of the original pair. Similar results are given for Jordan triple systems and algebras. Finally, we characterize the inner ideals of a simple pair which determine simple Ž subquotients, answering the question posed by O. Loos and E. Neher 1994, J.. Algebra 166, 255᎐295 . ᮊ

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Macdonald’s theorem for quadratic Jordan algebras

Cited by 20 publications

References 3 publications

Noncommutative Jordan Rings

Noncommutative Jordan Rings

Extending a Jordan Ring Homomorphism

Local and Subquotient Inheritance of Simplicity in Jordan Systems

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