Indecomposable and Isomorphic Objects in the Category of Monomial Matrices Over a Local Ring

We study the indecomposability and isomorphism of objects from the category of monomial matrices Mmat(K) over a commutative local principal ideal ring K (whose objects are square monomial matrices and the morphisms from X to Y are matrices C such that XC = CY). We also study the subcategory Mmat0(K) of the category Mmat(K) with the same objects and solely with morphisms that are monomial matrices.