We introduce a generalization of the Hermite normal form for matrices over a principal ideal ring R which may contain zero divisors. That normal form allows us to solve the following basic linear algebra problems. Equality decision, containment test and element test for submodules of R k , k 2 IN. The determination of images, kernels, and inverse images of homomorphisms ' : R l ! R k , l; k 2 I...

For discrete-time linear systems over a principal ideal domain three types of (C;A)-invariance can be distinguished. Connections between these notions are investigated. For pure submodules necessary and su cient conditions for dynamic (C;A)-injection invariance are given. Su cient conditions are obtained in the general case. Mathematical Subject Classi cations (1991): 93B07, 93B99, 15A33, 13C99

The main result of this paper states that if R is a right Noetherian right bounded prime ring such that nonzero prime ideals are maximal and such that every proper homomorphic image of R is a principal right ideal ring then R is right hereditary. In [10, Theorem 8] it is proved that if R is a right bounded prime ring of finite right Goldie dimension such that every proper homomorphic image is a...