Racine, Michel L. Minimal identities for Jordan algebras of degree 2. (English) Zbl 0579.17014 Commun. Algebra 13, 2493-2506 (1985). The purpose of the paper under review is the investigation of the polynomial identities satisfied by all Jordan algebras of degree 2 (i.e. for all algebras J(Q,1) obtained from a quadratic form Q). The main result claims that over a field K of characteristic zero all identities of minimal degree are consequences of \[ k=x^ 2\quad S_ 3(V(x_ 1),V(x_ 2),V(x_ 3))-x\quad S_ 3(V(x_ 1),V(x_ 2),V(x_ 3))\circ x, \] where \(S_ 3\) is the standard identity and \(zV(x)=z\circ x\). It is shown that the Jordan element of \(K<x,y,z>\lambda =[[x,y]^ 2,[x,z]]\) is an identity of J(Q,1) but \(\lambda =0\) does not follow from \(k=0\). The author states the conjecture that the polynomials k and \(\lambda\) form a basis for the T-ideal of J(Q,1). Some identities for the Jordan algebras of degree 3 are obtained as well. Reviewer: V.Drensky Cited in 11 Documents MSC: 17C05 Identities and free Jordan structures 16Rxx Rings with polynomial identity Keywords:polynomial identities; Jordan algebras of degree 2; characteristic zero; standard identity; T-ideal; Jordan algebras of degree 3 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] DOI: 10.1090/S0002-9939-1950-0036751-9 · doi:10.1090/S0002-9939-1950-0036751-9 [2] Leron U., Trans. A.M.S 183 pp 175– (1973) · doi:10.1090/S0002-9947-1973-0332873-3 [3] DOI: 10.1016/0021-8693(82)90216-2 · Zbl 0486.17008 · doi:10.1016/0021-8693(82)90216-2 [4] Racine M.L., J. Algebras 41 pp 224– (1976) · Zbl 0336.17007 · doi:10.1016/0021-8693(76)90179-4 [5] DOI: 10.1080/00927877708822165 · Zbl 0368.17007 · doi:10.1080/00927877708822165 [6] Drensky V.S., Algebra i Logika 20 (2) pp 282– (1980) [7] Drensky V.S., C.R. Acad. Bulgare Sci 35 (2) pp 1327– (1982) 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.