Additive maps preserving determinant on module of symmetric matrices over Zm

Abstract

In order to characterize the additive maps preserving of modulus of symmetric matrices over residue class rings, these maps are firstly proved to be linear in fact, then they are classified and discussed by means of contract transformation, number theory knowledge, determinant operation, and standard prime factorization of integers, to determine the image of the main base, and thus characterize the image of all matrices using the linearity. The relationship between the maps which have different forms but belong to the same class in fact is also discussed. The results show that additive maps preserving determinant on modulus of symmetric matrices over residue class rings are all trival. The research method solves the difficulty caused by the fact that non-zero elements in a general ring are not necessarily invertible, and extends the basic set to the residue class rings. This result can be regarded as a small step toward determinant preserving problem in a ring, which improves the existing results of the linear preserving problem. It has reference value for the study of other preserving problems on the remaining class rings.

Keywords

• linear algebra
• additive maps
• the residual class ring
• matrix module
• preserving determinant