Self-injective rings and linear (weak) inverses of linear finite automata over rings

Let R be a finite commutative ring with identity and tau be a nonnegative i nteger. In studying linear finite automats, one of the basic problems is ho w to characterize the class of rings which have the property that every (we akly) invertible linear finite automaton M with delay tau over R has a line ar finite automaton M' over R which is a (weak) inverse with delay tau of M . The rings and linear finite automata are studied by means of modules and it is proved that *-rings are equivalent to self-injective rings, and the u nsolved problem (for tau = 0) is solved. Moreover, a further problem of how to characterize the class of rings which have the property that every inve rtible with delay tau linear finite automaton M over R has a linear finite automaton M' over R which is an inverse with delay tau' for some tau' great er than or equal to tau is studied and solved.