Fuzzy nil radicals and fuzzy primary ideals (Q1180250)

A further investigation of \textit{W. Liu}'s [ibid. 8, 133-139 (1982; Zbl 0488.20005)] notions of a fuzzy subring and a fuzzy ideal in an ordinary ring is given. The adjective fuzzy stems from the fact that for a fuzzy subring it is not possible to determine which ring elements definitely belong to the fuzzy subring and which definitely do not. Here a profound analysis is given of the newly introduced notions of fuzzy nil radical of a fuzzy ideal in a ring and of a fuzzy primary ideal of a ring. A lot of interesting properties have been proved and where necessary illustrated by means of counterexamples. Finally the algebraic nature of direct and inverse images of fuzzy primary ideals under surjective homomorphisms is investigated. Although the author was aware about \textit{Y. Zhang}'s notion [ibid. 27, 345-350 (1988; Zbl 0663.13001)] of prime \(L\)-fuzzy ideal and primary \(L\)-fuzzy ideal, he did not outline the relation to his concepts.