On Additivity of Jordan Higher Mappings on Generalized Matrix Algebras

In this article, the additivity of higher multiplicative mappings, i.e., Jordan mappings, on generalized matrix algebras are studied. Also, the definition of Jordan higher triple product homomorphism is introduced and its additivity on generalized matrix algebras is studied.


Introduction
Suppose that a ring has identity and commutative, and and are associative algebras on . Let be -bi module and be -bi module. and are two bi module homomorphisms, where and , satisfying and for all and .
[ ] , with a usual matrix like multiplication, where either or is a generalized matrix algebra.
if is faithful as a left -module (resp.,rightmodule ) and is said to be a Triangular algebra [1]. See Cheng and Jing [2] for examples on nest algebras and block upper Triangular matrix algebras . For more examples, see Xiao and Wei [3].
Let : and . 1. is called multiplicative if . 2. is said to be a Jordan map if 3. is said to be a Jordan Triple Product Homomorphism if [1]. Characterizing the interrelation between the additive structures and the multiplicative of algebra or a ring is an interesting topic. This relation was first studied by Martindale [4] on a prime ring with some condition, and in [ 2,5,6 ] on operator algebras. Additivity of maps which are multiplicative

ISSN: 0067-2904
Shaheen Iraqi Journal of Science, 2021, Vol. 62, No. 4, pp: 1334-1343 1335 with respect to product on operator algebras, such that Jordan -triple product homomorphism, were investigated by Lu [ 7 ]. Ling and Lu [ 8 ] studied Jordan maps on nest algebras. They proved that any bijective Jordan map on standard sub algebra of nest algebra is an additive, which was extended to surjective Jordan pair maps of Triangular algebra by Ji [9] . Cheng and Jing [ 2] studied the linearity of multiplicative(Jordan)( triple) bijective map and elementary surjective map on triangular algebra. Li and Xiao [ 1], under some conditions, extended the results of Ji [9] to generalized matrix algebras. Shaheen [10 ] introduced the definition of higher multiplicative mapping and Jordan higher mapping and studied their additivity property on triangular matrix ring . In this article, we study the additivity property of them on generalized matrix algebras .
Li and Jing [ 11 ] studied the additive property of Jordan triple product homomorphism of prime ring . Kuzma [ 12 ] described Jordan triple product homomorphisms of matrix algebra . Motivated by the results of Li. and Jing [11 ] and Li. and Xiao [ 1 ], the authors in Kim and Park [ 13 ] studied the additivity of Jordan -Triple product homomorphism from generalized matrix algebras. For more results about Jordan triple homomorphism, see [14,15]. In this article , the definition of Jordan higher triple product homomorphism is introduced and its additivity on generalized matrix algebra is studied. We use techniques similar to those used by Lu [ 7 ] and Kim and Park [13]. Throughout this article, let = . We set

Then
, where for .

2-Additivity of Higher Multiplicative Mappings on Generalized Matrix Algebra
Definition 2.1 [ 10]. The family of mappings on a ring R is said to be

Theorem 2.2
Let be the generalized matrix algebra that satisfies the conditions: 1.
or .

4.
or n Then any bijective higher multiplicative mapping from onto arbitrary ring is additive. We shall introduce the following Lemmas to prove Theorem 2. .

Corollary 2.9:
Let be an algebra (unital, not necessarily prime) over a commutative ring . Then, any bijective higher multiplicative mapping from , onto the arbitrary ring is additive Proof: Clearly, we achieve the result when =0.

3-Additivity of Jordan Higher Maps on Generalized Matrix Algebra
In this section, we study the additivity of Jordan higher maps on generalized matrix algebra.