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

Let R be a finite commutative ring with identity and τ be a nonnegative integer. In studying linear finite automata, one of the basic problems is how to characterize the class of rings which have the property that every (weakly) invertible linear finite automaton μ with delay τ over R has a linear finite automaton ,μ’ over R which is a (weak) inverse with delay τ of μ. 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 unsolved problem (for τ = 0) is solved. Moreover, a further problem of how to characterize the class of rings which have the property that every invertible with delay r linear finite automaton ,μ over R has a linear finite automaton ,μ ’ over R which is an inverse with delay τ’ for some τ’≥ τ is studied and solved.

LetR be a finite commutative ring with identity and τ be a nonnegative integer. In studying linear finite automata, one of the basic problems is how to characterize the class of rings which have the property that every (weakly) invertible linear finite automaton ? with delay τ over R has a linear finite automaton ?′ over R which is a (weak) inverse with delay τ of ?. 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 unsolved problem (for τ=0) is solved. Moreover, a further problem of how to characterize the class of rings which have the property that every invertible with delay τ linear finite automaton ? overR has a linear finite automaton ?′ over R which is an inverse with delay τ′ for some τ′?τ is studied and solved.