Approximate Multiplicative Left Derivations On Rank-s Matrices

Ring theory is an important branch of algebra.All the time,derivations are an important class of tools in the course of ring theory study.In 1990,Bresar and Vukman came up with the concept of left derivations,and studied left derivations on prime rings and Banach algebras.In the 1940s,scholars have been concerned about the effect of additive conditions on derivations.In 1991,Daif studied the conditions that multiplicative derivations are additive(i.e.derivations).Naturally,we obtain multiplicative left derivations by removing the condition of additivity from left derivations.Hence the first purpose of this paper is describing multiplicative left derivations from full and upper triangular matrix rings over unital associative rings.On the other hand,in 2012,Franca introduced the idea of linear preserver problems into the theory of functional identities.All the time,linear preserver problems are mostly developed on the subsets that are not closed under addition.Affected by this,scholars have paid more and more attention to the properties of functional identities on nonadditive subsets of the ring of all n×n matrices,such as the set of all invertible matrices,the set of all singular matrices,and the set of all rank-s matrices.The second purpose of this paper is also under the background and research framework above,we research the approximate multiplicative left derivations on the set of all rank-s matrices,and the specific results are as follows.Firstly,let A be a unital associative ring,we describe multiplicative left derivations from full and upper triangular matrix rings over A,and we characterize multiplicative left derivations on some multiplicatively closed subsets of full matrix rings over A.Secondly,for the division ring D,we prove that an approximate multiplicative left derivation on the set of all rank-s matrices is equal to zero on the local scale,where 1≤s≤n is a fixed integer.