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Isomorphisms of simple and Artinian H-rings. (Italian. English summary) Zbl 0542.16038

Via a series of non-trivial lemmata the author achieves a handsome isomorphism characterization for simple artinian Hestenes ternary rings; she uses the classification of these rings given by F. Bartolozzi, G. Panella [Ric. Mat. 26, 255-275 (1977; Zbl 0402.17006)] and L. Profera [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 62, 292-299 (1977; Zbl 0377.16020)]: Let F be an sfield, V,W non-trivial finitely generated left vector spaces over F with \(n=\dim_ FV\), \(m=\dim_ FW\) and \(H_ 1=Hom_ F(V,W)\), \(H_ 2=Hom_ F(W,V)\). By F(n,m) the author denotes the Hestenes ternary ring with the additive group \(H_ 1\oplus H_ 2\) and the ternary \(operation\)
(\((p{}_ 1,p_ 2),(q_ 1,q_ 2),(t_ 1,t_ 2))\to(t_ 1q_ 2p_ 1,p_ 2q_ 1t_ 2)\) where \(p_ i,q_ it_ i\in H_ i\) for \(i=1,2.\)
In addition let \(\tau\) be an involutional anti-automorphism of F and h, g non-degenerated relative to \(\tau\) right sesquilinear forms on V, W, resp., both hermitian or both alternate. \(F(\tau\),n,m,h,g) denotes the Hestenes ternary ring with the additive group \(H_ 1\) and the ternary operation \((p,q,t)\mapsto pq^*t\) where \(p,q,t\in H_ 1\) and \(q^*\in H_ 2\) with \(h(v,wq^*)=g(vq,w)\) for \(v\in V\), \(w\in W\). As proven l.c. every simple artinian Hestenes ternary ring is isomorphic to one of the two kinds just mentioned and no ring of the first kind is isomorphic to one of the second. So the whole isomorphism problem for simple artinian Hestenes ternary rings is solved by the following theorems:
\(F(n,m)\simeq F'(n',m')\) iff \(n=n'\), \(m=m'\) and F is isomorphic or anti- isomorphic to F’. \(F(\tau,n,m,h,g)\simeq F'(\tau ',n',m',h',g')\) iff \(n=n'\), \(m=m'\) and there are \(\phi^{-1}\)-semilinear bijections \(\mu\) :V’\(\to V\), \(\rho\) :W’\(\to W\) with an isomorphism \(\phi\) of F onto F’ and \(c\in F'\backslash \{0\}\) such that \(h'(x,y)=h(x\mu,y\mu)^{\phi}c\) (x,\(y\in V')\) and \(g'(x,y)=g(x\rho,y\rho)^{\phi}c\) (x,\(y\in W')\).
Reviewer: W.Lex

MSC:

16Y60 Semirings
17A40 Ternary compositions
15A63 Quadratic and bilinear forms, inner products
16Kxx Division rings and semisimple Artin rings
16S50 Endomorphism rings; matrix rings
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