A Dice Rolling Game on a Set of Tori

Abstract

Some linear algebraic and combinatorial problems are widely studied in connection with $\sigma$-games. One particular
issue is to characterize whether or not a given vector lies in the submodule generated by the rows of a given matrix over a commutative ring. In general, one can solve this problem easily and algorithmically using the linear algebra over commutative ring. However, if the matrix has some combinatorial structure, one may expect that some more can be asserted instead of merely giving an algorithm. A recent outstanding example appeared in this line of research is the paper by Florence and Meunier published in Journal of Algebraic Combinatorics in 2010. In the same spirit, we consider a matrix over  $\mathbb{Z}_n$ to completely characterize the submodule generated by its rows and give a constructive proof. The main idea for the characterization is to find certain good basic elements in the row space and then express a given element as the linear combination of them as well as some additional term.