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The goal of the present article is to extend the study of commutative rings whose ideals form an MV-algebra as carried out by Belluce and Di Nola [1] to non-commutative rings. We study and characterize all rings whose ideals form a pseudo MV-algebra, which shall be called here generalized Lukasiewicz rings. We obtain that these are (up to isomorphism) exactly the direct sums of unitary special primary rings.

Residuated lattices play an important role in the study of fuzzy logic based ont-norms. In this paper, we introduce some notions ofn-fold filters in residuated lattices, study the relations among them, and compare them with prime, maximal and primary, filters. This work generalizes existing results in BL-algebras and residuated lattices, most notably the works of Lele et al., Motamed et al., Haveski et al., Borzooei et al., Van Gasse et al., Kondo et al., Turunen et al., and Borumand Saeid et al., we draw diagrams summarizing the relations between different types ofn-fold filters andn-fold residuated lattices.

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Given a pseudoresiduated latticeMand a latticeL, we introduce and characterize the fuzzy versions of differentn-fold implicative (resp., obstinate, Boolean, normal, and extended involutive) filters ofM. Moreover, we study some relationships between these different types of fuzzyn-fold filters.