Multi-mappings On Associative Rings

Homomorphisms and derivations are basic and important mappings on rings. Jordanhomomorphisms and multi-derivations have been researched extensively. In this disser-tation, we consider multi-mappings on rings. On one hand, we introduce notions of Jor-dan multi-homomorphisms, multi-homomorphisms and multi-anti-homomorphisms. Us-ing Boolean homomorphisms, we describe the structure of Jordan multi-homomorphisms,and then show that concepts of Jordan multi-homomorphisms, multi-homomorphismsand multi-anti-homomorphisms are the same. Meanwhile, general forms of Boolean ho-momorphisms on some special rings are given. On the other hand, we introduce conceptsof (n, m)-derivation-homomorphisms and Boolean n-derivations. Using n-derivations andm-homomorphisms, we describe the structure of (n, m)-derivation-homomorphisms.Let R1,..., Rn, R and S be rings. Unless otherwise specified, the word “ring”throughout this dissertation shall mean a ring with an identity element.In Chapter1, we provide a brief survey about Jordan homomorphisms and multi-derivations. Meanwhile, basic concepts and notations involved in this dissertation aregiven.In Chapter2, we introduce notions of Jordan multi-homomorphisms, multi-homomo-rphisms and multi-anti-homomorphisms. A Jordan n-homomorphism can be consideredas a mapping such that it is a homomorphism for each variable while the others areleft fixed. Meanwhile, n-homomorphisms and n-anti-homomorphisms can be consideredsimilarly.Definition2.1.2A Jordan n-homomorphism f: R×···×R�'S is said to be permuting, if the equation f(a1,..., an)=f(aσ(1),...,aσ(n))holds for all a1,...,an∈R and for every permutation {σ(1),...,σ(n)}. A permuting Jordan2-homomorphism is also called a symmetric Jordan bi-homomorphism.Definition2.1.3A homomorphism Φ:R�'S is said to be a Boolean homomor-phism, if the equation Φ(x)=Φ(x)2holds for all x∈R.Definition2.1.5Let Ri, S be rings, Φi: Ri�'S be mappings,i=1,2.(1)Φ1,Φ2are said to be commutative, if the equation Φ1(a)Φ2(b)=Φ2(b)Φ1(a) holds for all (a,b)∈R1×R2.(2)Φ1,Φ2are said to be orthogonal, if the equation Φ1(a)Φ2(b)Φ2(b)Φ1(a)=0holds for all (a, b)∈R1×R2. Using Boolean homomorphism, we describe the structure of Jordan bi-homomorphi-sms and get the following results.Theorem2.1.1Let f be a mapping from R1×R2to S. Then f is a Jordan bi-homomorphism if and only if there exist a pair of commutative Boolean homomorphisms0: R1�'S and ψ: R2�'S such that f=Φ*ψ. Furthermore, if Φ(1)=ψ(1), f has a unique decomposition.Through the structure of Jordan bi-homomorphism, we describes the structure of Jordan n-homomorphisms and get following results.Corollary2.1.1Let f be a mapping from R x R to S. Then f is a symmetric Jordan bi-homomorphism if and only if there exists a unique Boolean homomorphism Φ: R�'S such that f=Φ*Φ.Theorem2.1.2Let f be a mapping from R1×…×Rn to S. Then f is a Jordan n-homomorphism if and only if there exist pairwise commutative Boolean homomorphisms Φi: Ri�'S for i∈{1,...,n} such that f=Φ1… Φn. Furthermore, if R1=…=Rn f has a unique decomposition.Corollary2.1.2Let f be a mapping from R×…×R to S. Then f is a permuting Jordan n-homomorphism if and only if there exists a unique Boolean homomorphism0: R�'S such that f=Φ*…*Φ.After describing the structure of Jordan multi-homomorphisms, we show that con-cepts of Jordan multi-homomorphisms, multi-homomorphisms and multi-anti-homomor-phisms are the same.Theorem2.2.1Let f be a mapping from R1×…×Rn to S. Then f is a Jordan n-homomorphism if and only if f is an n-homomorphism if and only if f is an n-anti- homomorphism.In Chapter3, we introduce notions of derivation-homomorphisms and Boolean deriva-tions. A derivation-homomorphism can be considered as a mapping, such that it is a derivation for the first variable while the second variable is left fixed, and it is a homo-morphism for the second variable while the first variable is left fixed.Definition3.1.2A derivation?: R�'S is said to be a Boolean derivation, if the equation?(x)=?(x)2holds for all x∈R.Using Boolean derivations we describe the structure of derivation-homomorphisms and get the following theorem.Theorem3.1.1Let f be a derivation-homomorphism from R1×R2to S and Z(S) be the center of S. If the image of R1×{1} has an identity element, then there exist a Boolean derivation?: R1�'S and a homomorphism A: R2�'Z(S) such that/=?*λ. Furthermore, if the identity element of S has an inverse image, then/has a unique decomposition.By considering general forms of derivation-homomorphisms on special rings, we get the following theorem.Theorem3.1.2Let S be a semiprime ring. Then any derivation-homomorphism from R1×R2to S must be zero.Then we introduce concepts of (n, m)-derivation-homomorphisms and Boolean n-derivations. A (n, m)-derivation-homomorphism can be considered as a mapping, such that it is a derivation or homomorphism for each variable while the others are left fixed.Definition3.2.2A n-derivation?: R1×…×Rn�'S is said to be a Boolean n-derivation, if the equation?(x1,..., xn)=?(x1,..., xn)2holds for all (x1,..., xn)∈0R1×…×Rn.Using n-derivations and m-homomorphisms, we describe the structure of (n, m)-derivation-homomorphisms.Theorem3.2.1Let f be an (n, m)-derivation-homomorphism from R1×…×Rn to S. If the identity element of S has an inverse image, then there exist a unique Boolean n-derivation?: R1×…×Rn�'S and a unique m-homomorphism A: Rn+1×…×Rn+m�'Z(S) such that/=?*Aλ...