Research On Related Problems Of L-fuzzy Subrings

In this paper,the fuzzy subring,fuzzy ideal,fuzzy quotient ring and fuzzy prime ideal in fuzzy algebra are extended theoretically.The definitions of L-fuzzy subring degree,L-fuzzy ideal degree,L-fuzzy quotient ring degree and L-fuzzy prime ideal degree are given,and the equivalent characterizations of their definitions are given respectively;The relationship between L-fuzzy subring degree,L-fuzzy ideal degree,L-fuzzy quotient ring degree and L-fuzzy convex structure is also discussed;The related properties of these concepts are studied.The research content of this paper mainly includes four parts.When L is a completely distributive lattice,the specific contents are as follows:In the first part,the definition of L-fuzzy subring degree is given,and the equivalent characterization theorem of L-fuzzy subring degree is discussed.On this basis,the relationship between L-fuzzy subring degree and L-fuzzy convex structure is discussed.It is pointed out that ring homomorphism can be regarded as L-fuzzy convex preserving mapping and L-fuzzy convex to convex mapping.The relationship between the subring degree of L-fuzzy subset of ring and the L-fuzzy subring degree of image and original image of L-fuzzy subset under homomorphic mapping is discussed.In addition,the representation theorems of inner product and direct sum of L-fuzzy subrings are given,and the degree of L-fuzzy subrings of inner product,outer product and direct sum of L-fuzzy subsets are further discussed.In the second part,the definition and equivalent characterization of L-fuzzy ideal degree are given.At the same time,the relationship between L-fuzzy ideal degree and L-fuzzy convex structure,and the relationship between the ideal degree of the L-fuzzy subset of the ring and the L-fuzzy ideal degree of the image and the original image of the L-fuzzy subset under homomorphic mapping are discussed.In the third part,when the subring A of ring R is fuzzy and the ideal N is clear,the definition of L-fuzzy quotient ring degree is given,and the relationship between L-fuzzy quotient ring degree and L-fuzzy convex structure is discussed.At the same time,the relationship between L-fuzzy subring degree and L-fuzzy quotient ring degree is studied.It is proposed that the ring homomorphism from ring to quotient ring can be regarded as Lfuzzy convex to convex mapping and L-fuzzy convex preserving mapping.In particular,when the degree of L-fuzzy quotient ring is 1,L-fuzzy quotient ring is characterized by representation theorem.The fourth part gives the definition and equivalent characterization of L-fuzzy prime ideal degree,and the relationship between the prime ideal degree of the L-fuzzy subset of the ring and the L-fuzzy prime ideal degree of the image and the original image of the L-fuzzy subset under homomorphic mapping are discussed.