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

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Abstract

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 ℳ with delay τ over R has a linear finite automaton ℳ′ over R which is a (weak) inverse with delay τ 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 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 ℳ overR has a linear finite automaton ℳ′ over R which is an inverse with delay τ′ for some τ′⩾τ is studied and solved.

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Authors and Affiliations

  1. State Key Laboratory of Information Security, Graduate School, University of Science and Technology of China, 100039, Beijing, China

    Haiwen Ou & Zongduo Dai

Authors
  1. Haiwen Ou
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  2. Zongduo Dai
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Additional information

Project supported by the National Natural Science Foundation of China(Grant No. 69773015).

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Ou, H., Dai, Z. Self-injective rings and linear (weak) inverses of linear finite automata over rings. Sci. China Ser. A-Math. 42, 140–146 (1999). https://doi.org/10.1007/BF02876565

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  • DOI: https://doi.org/10.1007/BF02876565

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