Abstract
This paper extends the study of commutative rings whose ideals form an MV-algebra as carried out by Belluce and Di Nola (Math Log Q 55(5):468–486, 2009) to non-commutative rings. We study and characterize all rings whose ideals form a pseudo-MV-algebra, which shall be called here generalized Łukasiewicz rings. We obtain that up to isomorphism, these are exactly the direct sums of unitary special primary rings.
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References
Belluce LP, Di Nola A (2009) Commutative rings whose ideals form an MV-algebra. Math Log Q 55(5):468–486
Chajda I, Khür J (2006) GMV-algebras and meet-semilattices with sectionally antitone permutations. Math Slovaca 56:275–288
Cignoli R, D’Ottaviano I, Mundici D (2000) Algebraic foundations of many-valued reasoning. Kluwer, Dordrecht
Cignoli R, Mundici D (1998) An elementary presentation of the equivalence between MV-algebras and \(\ell \)-groups with strong units. Stud Log 61:49–64 special issue on many-valued logics
Dvurečenskij A (2001) On pseudo MV-algebras. Soft Comput 5(5):347–354
Dvurečenskij A (2002) Pseudo MV-algebras are intervals in \(\ell \)-groups. J Aust Math Soc 72:427–445
Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer, Ister Science, Dordrecht, Bratislava
Georgescu G, Iorgulescu A (1999) Pseudo-MV algebras: a non-commutative extension of MV algebras. In: Proc 4th International Symposium of Economics Informatics. IN-FOREC Printing House, Bucharest, pp 961–968
Georgescu G, Iorgulescu A (2001) Pseudo-MV algebras I. Mult Valued Log 6:95–135
Griffin M (1974) Multiplication rings via their total quotient rings. Can J Math 26:430–449
Hajarnavis CR, Norton NC (1985) On dual ring and their modules. J Algebra 93:253–266
Hájek P (1998) Metamathematics of fuzzy logic. Kluwer, Dordrecht
Kaplansky I (1948) Dual rings. Ann Math 49:689–701
Lam TY (1991) A first course in non-commutative rings. Springer, New York
Wadsworth AR (1987) Dubrovin valuation rings. In: Van Oystaeyen F, Le Bruyn L (eds) Perspectives in ring theory. Springer, Netherlands, pp 359–374
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Communicated by Y. Yang.
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Kadji, A., Lele, C. & Nganou, J.B. Generalized Łukasiewicz rings. Soft Comput 21, 2469–2476 (2017). https://doi.org/10.1007/s00500-017-2575-x
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DOI: https://doi.org/10.1007/s00500-017-2575-x