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Formalization of Ring Theory in PVS

Isomorphism Theorems, Principal, Prime and Maximal Ideals, Chinese Remainder Theorem

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Abstract

This paper presents a PVS development of relevant results of the theory of rings. The PVS theory includes complete proofs of the three classical isomorphism theorems for rings, and characterizations of principal, prime and maximal ideals. Algebraic concepts and properties are specified and formalized as generally as possible allowing in this manner their application to other algebraic structures. The development provides the required elements to formalize important algebraic theorems. In particular, the paper presents the formalization of the general algebraic-theoretical version of the Chinese remainder theorem (CRT) for the theory of rings, as given in abstract algebra textbooks, proved as a consequence of the first isomorphism theorem. Also, the PVS theory includes a formalization of the number-theoretical version of CRT for the structure of integers, which is the version of CRT found in formalizations. CRT for integers is obtained as a consequence of the general version of CRT for the theory of rings.

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  1. Available at https://shemesh.larc.nasa.gov/fm/ftp/larc/PVS-library/.

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Authors and Affiliations

  1. Universidade Federal de Goiás, Goiânia, Brazil

    Thaynara Arielly de Lima

  2. Universidade Federal de Catalão, Catalão, Brazil

    André Luiz Galdino

  3. Universidade de Brasília, Brasília, Brazil

    Andréia Borges Avelar & Mauricio Ayala-Rincón

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  1. Thaynara Arielly de Lima
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  2. André Luiz Galdino
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  3. Andréia Borges Avelar
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  4. Mauricio Ayala-Rincón
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Correspondence to Mauricio Ayala-Rincón.

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Research funded by FAPDF Grant Ed. DE 03/2018, 135/2019. Fourth author partially funded by CNPq Grant 07672/2017-4.

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de Lima, T.A., Galdino, A.L., Avelar, A.B. et al. Formalization of Ring Theory in PVS. J Autom Reasoning 65, 1231–1263 (2021). https://doi.org/10.1007/s10817-021-09593-0

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