Rings with chain conditions (Q2277554)

This paper concerns the concept of a fuzzy ideal in a classical ring as introduced by \textit{W.-J. Liu} [in Fuzzy Sets Syst. 8, 133-139 (1982; Zbl 0488.20005)]. The paper focusses on characterizations of left Artinian and left Noetherian rings in terms of fuzzy ideals. More explicitly, it is proved that a ring with unity is left Artinian iff every fuzzy left ideal on it has a finite range. A ring is left Noetherian iff every fuzzy left ideal on it has a well-ordered range. Using the characterization of a left Artinian ring, the authors prove that the cartesian product of two left Artinian rings with unity again constitutes a left Artinian ring. Finally an alternative proof of Hilbert's basis theorem is given.