Derivations with invertible values in flexible algebras
DOI:
https://doi.org/10.5269/bspm.v38i6.37192Keywords:
Derivations, invertible values, flexible algebras, Cayley-Dickson algebrasAbstract
Derivations with invertible values of 0 – torsion flexible algebras satisfying x(yz) = (xz)y over an algebraically closed field are described. For this class of algebra with unit element 1 and derivation with invertible value d is either a Cayley – Dickson algebra over its center Z(A) or a factor algebra of polynomial algebra C[a]/(a2) over a Cayley – Dickson division algebra; also C is 2 – torsion, d(C) = 0 and d(a) = 1+ua for some u in center of C and d is an outer derivation. Moreover, C is a split Cayley – Dickson algebra over its center Z having a derivation with invertible value d if and only if C is obtained by means of Cayley – Dickson process from its associative division subalgebra and can be represented as a direct sum C = V ⊕ aV.
References
2. J. Bergen, I. Herstein and C. Lanski,Derivations with invertible values, Canad. J. Math 35(1983), 300-310.
3. J. Bergen and L. Carini, Derivations with invertible values on a Lie ideal, Canad. Math. Bull 31(1988), 103-110.
4. J.C. Chang, α-derivations with invertible values, Bull. Inst. Math. Acad. Sinica 13(1985), 323-333.
5. C. Demir, E. Albas, N. Argac and A. Fosner, Superderivations with invertible values, J. Algebra Appl 14(2015), 11pp.
6. A. Elduque and H.C. Myung, On flexible Composition algebras, Comm. Algebra 21(1993), 2481-2505.
7. A. Elduque and H.C. Myung, Flexible Composition algebras and Okubo algebras, Comm. Algebra 19(1991), 1197-1227.
8. A. Giambruno, P. Misso and P.C. Milies, Derivations with invertible values in rings with involution, Pac.J. Math 123(1986), 47-54.
9. M. Hongan and H. Komatsu, (σ, τ)-Derivations with invertible values, Bull. Inst. Math. Acad. Sinica 15(1987), 411-415.
10. I. Kaygorodov and Y. Popov,Alternative algebras admitting derivations with invertible values and invertible derivations, Izvestiya. Math 78(2014), 922-935.
11. I. Kaygorodov, A. Lopatin and Y. Popov, Jordan algebras admitting derivations with invertible values, arXiv: 1511.00742.
12. H. Komatsu and A. Nakajima, Generalized derivations with invertible values, Comm. Algebra 32(2004), 1937-1944.
13. A.A. Popov,Differentiably simple alternative algebras, Algebra and Logic 49(2010), 456-469.
14. K.A. Zhevlakov, A.M. Slinko, I.P. Shestakov and A.I. Shirshov, Rings that are nearly associative, Pure and Applied Mathematics, 104, Academic Press, Inc, New York-London, 1982.
Downloads
Published
Issue
Section
License
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
