# Maps preserving triple product on rings

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, P.O.Box 47416-1468, Babolsar, Iran.

10.22103/jmmrc.2021.16645.1124

Abstract

Let R and R0 be two unital rings such that R contains a non-
trivial idempotent P1. If R is a prime ring, we characterize the form of bijective
map ' : R ! R0 which satis es '(ABP) = '(A)'(B)'(P), for every A;B 2 R
and P 2 fP1; P2g, where P2 := I 􀀀 P1 and I is the unit member of R. It is
shown that ' is an isomorphism multiplied by a central element. Finally, we
characterize the form of ' : R ! R which satis es '(P)'(A)'(P) = PAP,
for every A 2 R and P 2 fP1; P2g.

Keywords

#### References

V. Darvish, N.M. Nazari, H. Rohi, A. Taghavi, Maps preserving $eta-$product $AP+eta PA^*$ on $C^*-$algebras, J. Korean Math. Soc. 54 (3), 867-876 (2017).
H. Gao, $*-$Jordan-triple multiplicative surjective maps on $B(H)$, J. Math. Anal. Appl. 401, 397-403 (2013).
L. Gonga, X. Qi, J. Shao, F. Zhang, Strong (skew) $xi-$Lie commutativity preserving maps on algebras, Cogent Math. 2(1), 1003175 (2015).
C. Jianlian, C. Park, Maps preserving strong skew lie product on factor von Neumann algebras, Acta Math. Scientia. 32(2), 531-538 (2012).
P. Li, F. Lu, Nonlinear maps preserving the Jordan triple 1-$*$-product on von Neumann algebras, Complex Anal. Oper. Theory. 11, 109–117 (2017).
L. Liu, LG.X. Ji, Maps preserving Product $X^*Y+ YX^*$ on factor von Neumann algebras, Linear Multilinear Algebra. 59(9), 951-955 (2011).
W.S. Martindale III, When are multiplicative mappings additive?, Proc. Amer. Math. Soc. 21, 695-698 (1969).
L. Moln'{a}r, On isomorphisms of standard operator algebras, Studia Math.  142(3), 295-302 (2000).
L. Moln'{a}r, Multiplicative Jordan triple isomorphisms on the self-adjoint elements of von Neumann algebras, Linear Algebra Appl. 419, 586-600 (2006).
L. Moln'{a}r, P. '{S}emrl, P., Transformations of the unitary group on a Hilbert space, J. Math. Anal. Appl. 388, 1205-1217 (2012).
A. Taghavi, Maps preserving Jordan triple product on the self-adjoint elements of $C^*-$algebras, Asian-European J. Math. 10, No. 2, 1750022(7 pages) (2017). 11, 391-405 (2020).
A. Taghavi, M. Razeghi, M. Nouri, V. Darvish, Maps preserving triple product $A^*B + BA^*$ on $*-$algebras, Asian-European J. Math. 12, No. 3, 1950038(13 pages) (2019).
A. Taghavi, S. Salehi, Continuous maps preserving Jordan triple products from $mathbb{GL}_1$ to $mathbb{GL}_2$ and $mathbb{GL}_3$, Linear Multilinear Algebra. Published online: 20 Mar (2019).
A. Taghavi, S. Salehi, Continuous maps preserving Jordan triple products from $mathbb{GL}_n(mathbb{C})$into $mathbb{C}^*$, Indagationes Mathematicae. 28, 1233-1239 (2017).
A. Taghavi, S. Salehi, Continuous maps preserving Jordan triple products from $mathbb{U}_n$ into $mathbb{D}_m$, Indagationes Mathematicae. 30, 157–164 (2019).
A. Taghavi, E. Tavakoli, Additivity of maps preserving Jordan triple products on prime $C^*-$algebras, Annals of Functional Analysis. 11, 391-405 (2020).
J.H. Zhang, F.J. Zhang, Nonlinear maps preserving Lie products on factor von Neumann algebra, Linear Algebra Appl. 42, 18-30 (2008).

### History

• Receive Date: 23 October 2020
• Revise Date: 26 December 2020
• Accept Date: 15 April 2021