Expanders on matrices over a finite chain ring, I

In this work and its sequel, we study the expanding phenomenon of matrices over a finite chain ring of large residue field. A sum-product estimate is proved. It is showed that $x+yz$ is a moderate expander on $n\times n$ matrices with exponent $\frac{n+1}{6}$. These results generalise the main theorems in a recent work of Xie and Ge. The proofs use spectral graph theory and elementary divisor theory.