Let L m denote the chain { 0 , 1 , 2 , … , m - 1 } with the usual ordering and M n ( L m ) the matrix semiring of all n × n matrices with elements in L m . We firstly introduce some order-preserving semiring homomorphisms from M n ( L m ) to M ( L k ) . By using these homomorphisms, we show that a matrix over the finite chain L m can be decomposed into the sum of some matrices over the finite chain L k , where k < m . As a result, cut matrices decomposition theorem of a fuzzy matrix (Theorem 4 in [Z.T. Fan, Q.S. Cheng, A survey on the powers of fuzzy matrices and FBAMs, International Journal of Computational Cognition 2 (2004) 1–25 (invited paper)]) is generalized and extended. Further, we study the index and periodicity of a matrix over a finite chain and get some new results. On the other hand, we introduce a semiring embedding mapping from the semiring M n ( L m ) to the direct product of the h copies of the semiring M n ( L k ) and discuss Green’s relations on the multiplicative semigroup of the semiring M n ( L m ) . We think that some results obtained in this paper is useful for the study of fuzzy matrices.