Asian Journal of Algebra 1994-540x 2077-2025 Asian Network for Scientific Information 10.3923/aja.2017.1.9 Lakshmi Devi G. Jayalakshmi K. 1 2017 10 1 Background: First examples of simple nonassociative superalgebras were constructed by Shestakov in (1991 and 1992). Since then many researchers showed interest towards the study of superalgebras and superalgebras of vector type. Materials and Methods: Multiplication in M is uniquely defined by a fixed finite set of derivations and by elements of A. The types of derivations used in this article to obtain the results are the near derivation δx,y : a ↦ (a, x, y) the derivation D : a ↦ (x, a, x) and the derivation Dij : a ↦ (xi ,a, xj) Results: The flexible Lie-admissible superalgebra FFLSA[φ; x] over a 2, 3-torsion free field Φ on one odd generator e is isomorphic to the twisted superalgebra B0 (Φ[Γ], D, γ0) with the free generator . In a 2, 3-torsion free flexible Lie-admissible superalgebras of vector type F, the even part A is differentiably simple, associative and commutative algebra and the odd part M is a finitely generated associative and commutative A-bimodule. Conclusion: A connection between the integral domains, the finitely generated projective modules over them, the derivations of an integral domain and the flexible Lie-admissible superalgebras of vector type has been established. If A is an integral domain and M = Ax1+…+Axn be a finitely generated projective A-module of rank 1, then F (A, Δ, Γ) is a flexible Lie-admissible superalgebra with even part A and odd part M provided that the mapping M = Axi+…+Axn is a nonzero derivation of A into the A-module (M⊗A M)*, Δ = {Dij |i, j = 1,…, n} is a set of derivations of A where Dij (a) = ā (x⊗xj).

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Okubo, S. and H.C. Myung, 1981 Trans. Am. Math. Soc., 264 459 472 Benkart, G.M. and J.M. Osborn, 1981 J. Algebra, 71 11 31 Kamiya, N. and S. Okubo, 2012 2 loop Algebra and tetrahedron Algebra.]]> Survey Note, 1809 42 53 Shestakov, I.P., 1991 Sib. Math. J., 32 1052 1060 Shestakov, I.P., 1998 Algebra Logic., 37 411 422 Zhelyabin, V.N., 2002 Algebra Logic, 41 152 172 Zhelyabin, V.N. and I.P. Shestakov, 2004 Sib. Math. J., 45 860 882 Zhelyabin, V.N., 2012 Sib. Math. J., 53 450 460 Zhelyabin, V.N., 2015 Sib. Math. J., 56 411 424 Anderson, C.T. and D.L. Outcult, 1968 J. Algebra, 10 310 320 Zhevlakov, K.A., A.M. Slinkov, I.P. Shestakov and A.I. Shirshov, 1982 Rings that are Nearly Associative. Pages: 371 Pages: 371 Shestakov, I.P., 1997 Algebra Logic., 36 389 412 Posner, E.C., 1960 Proc. Am. Math. Soc., 11 337 343 Yuan, S., 1964 Duke Math. J., 31 623 630 Pchelintsev, S.V. and I.P. Shestakov, 2015 J. Algebra, 423 54 86 Pchelintsev, S.V., 2010 Ablegra Logic, 49 262 288