Congruence Ring Soft And Fuzzy Soft Congruence

The ring theory is one of the most important algebraic structure. Fuzzy sets and soft sets are powerful tools for analyzing and dealing with uncertainty problems. Cross-study of the ring theory, fuzzy algebra and information science, not only enriches the theory of fuzzy algebra, but also provides a profound applications background for the ring theory. In recent years, combination of soft set theory and algebraic structure gives fresh ideas from algebra to the research on soft set theory, which leads great mass study on soft algebra system. Based on this, we generalize congruence relations and fuzzy congruences, define soft congruence relations and fuzzy soft congruences over rings, and discuss several related properties. Specific work is as follows:1. Combined soft sets with congruence relations, the concept of soft congru-cnce relations over rings is given, connections among soft congruence relations, congruence relations and fuzzy congruences are studied. Also, a one-to-one cor-respondence between idealistic soft rings and soft congruence relations is inves-tigated. Furthermore, soft isomorphism theorem is established.2. Base on soft sets, we generalize fuzzy congruences, proposed fuzzy soft congruences over rings, and investigate equivalent characterizations among fuzzy soft congruences, fuzzy congruences and soft congruence relations, which further describes the relationships among fuzzy soft sets, fuzzy sets and soft sets.3. Some related algebraic properties of fuzzy soft congruences under fuzzy soft operations are studied. At the same, the set of all fuzzy soft congruences associated with some fuzzy soft operations over a ring can form a complete lattice.4. With the homomorphism of two crisp rings, the image and the inverse image of fuzzy soft congruences are studied respectively, that is to say, the image and the inverse image of fuzzy soft congruences are still fuzzy soft congruences.5. The notion of idealistic fuzzy soft rings is introduced and a one-to-one correspondence between fuzzy soft congruences and idealistic fuzzy soft rings over rings is obtained. Moreover, a one-to-one correspondence between the set of all fuzzy soft congruences and all idealistic fuzzy soft rings associated is also investigated.6. The notion of fuzzy soft homomorphisms of fuzzy soft rings is given and fuzzy soft quotient rings over rings are established. Moreover, fuzzy soft homomorphism(isomorphism) theorems are studied.