Jordan Derivations On Full Matrix Rings

Derivation is an important topic in the research of ring theory. Matrix ring theory over associative ring is also an important subject in algebra. In 1995, Jφnrup studied the structure of derivations on M_n(R), and proved that: every derivation on M_n(R) is the sum of an inner derivation and a derivation induced by a derivation of R. In 2001, Cui Shihua extended the result in her thesis, proved that the derivation from some subrings of M _n(R) to M_n(R) have the similar structure. She also studied the derivation of infinite matrices rings, depicted the structure of derivations from finite-column matrix ring to M_n(R). In 2003, Wang Junlin studied the structure of derivation from the subring constituted by diagonal matrix of M_n(R) toM_n(R).Jordan derivation is a generalization of derivation and derivation is a special case of Jordan derivation. In the paper, we study the derivations from a subring to the ring of M_n(R), where R is a unital associative ring. The case of infinite matrix rings is also studied.Let R be a unital associative ring, M_n(R) be the ring of all n×n matrices over R,M_∞(R) be the set of infinite matrices on R, S and T be the subrings of M_∞(R) constituted by finite-column matrix and finite-row matrix in M_∞(R), respectively. A denotes the subring of M_∞(R) constituted by the matrices with only a finite number of non-zero elements, A' denotes the subring constituted by the diagonal matrices in A, andThe main results of this thesis are as follows.Theorem 3.1 Let R a unital associative ing, (?) = {ae_kk | a∈R} (1≤k≤n), and d be a Jordan derivation form (?) to M_n(R). Then there are unique Jordan derivationδ_k of R andsuch thatTheorem 3.2 Let R be a unital associative ing, d is a Jordan derivation from T_t(R) to M_n(R) (1≤t≤n). Then there are Jordan derivationsδ₁,...,δ_t, andsuch that for any diag(a₁,…, a_t, 0,…, 0)∈T_t(R),Theorem 3.3 Let R be a unital associative ing, d is a Jordan derivation from S_n(R),to M_n(R). Then there are Jordan derivationsδ_ij, i,j = 1,…, n, such that for any aE_n∈S_n(R),d(aE_n) = (a^δ_ij)_n×n.Theorem 3.4 Let R be a unital associative ing, d is a Jordan derivation from S_t(R) (1≤t < n) to M_n(R). Then there are Jordan derivationsδ_ij, i, j = 1,…, n, of R andsuch that for any ae₁₁ +…+ ae_tt∈S_t(R),Theorem 4.1 Let R be a unital associative ing, d is a Jordan derivation from A' to A. Then there are B∈S∩T and Jordan derivationδ_i of R such that for any X∈A',Theorem 4.2 Let R be a unital associative ing, d is a Jordan derivation from A_t to A. Then there are Jordan derivationsδ_ij, i,j= l,…,t,of R andsuch that for any ae_tt+…+ ae_tt∈A_t,...