In this paper, the concept of fuzzy normed ring is introduced and some basic properties related to it are established. Our definition of normed rings on fuzzy sets leads to a new structure, which we call a fuzzy normed ring. We define fuzzy normed ring homomorphism, fuzzy normed subring, fuzzy normed ideal, fuzzy normed prime ideal, and fuzzy normed maximal ideal of a normed ring, respectively. We show some algebraic properties of normed ring theory on fuzzy sets, prove theorems, and give relevant examples.

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In this study, we take M to be a monoid and we let ρ be an equivalence relation on M such that ρ is a congruence. So, ρ is a submonoid of the direct product of monoids M×M, and M/ρ={xρ:x∈M} is a monoid with the operation (xρ)(yρ)=(xy)ρ. First, we propose and prove an introductory lemma and we give a relevant example. Then, we show that if ρ can be presented by a finite complete rewriting system, then so can M. As the final part of our main result, we prove that if ρ can be presented by a finite complete rewriting system, then so can M/ρ.