The Homomorphism Theorems of M-Hazy Rings and Their Induced Fuzzifying Convexities

Abstract

In traditional ring theory, homomorphisms play a vital role in studying the relation between two algebraic structures. Homomorphism is essential for group theory and ring theory, just as continuous functions are important for topology and rigid movements in geometry. In this article, we propose fundamental theorems of homomorphisms of M-hazy rings. We also discuss the relation between M-hazy rings and M-hazy ideals. Some important results of M-hazy ring homomorphisms are studied. In recent years, convexity theory has become a helpful mathematical tool for studying extremum problems. Finally, M-fuzzifying convex spaces are induced by M-hazy rings.

A Novel Approach to the Fuzzification of Fields

“…For example, Li J. and Shi F.-G. first discovered the close relationship between fuzzy sublattice and fuzzy convex structures [10]. Afterwards, Liu and Shi applied (fuzzifying) convexities into Mhazy lattices [11], and M-fuzzifying groups [12], Mehmood and Shi applied (fuzzifying) convexities into M-hazzy rings [13,14] and An and Shi applied L-fuzzy convexities into L-fuzzy rings [15]. However, the research of the relationship between L-fuzzy convexity and L-fuzzy subfields are hardly available.…”

A Generalized Definition of Fuzzy Subrings

“…e notions of M-fuzzifying convex structure and (L, M)-fuzzy convex structure are introduced by Shi and Xiu in [9,10]. Actually, fuzzy convexity exists in many mathematical research areas, such as fuzzy vector spaces, fuzzy groups, fuzzy lattices, and fuzzy topologies (see [7,8,[11][12][13][14][15][16][17][18][19][20][21][22][23][24]).…”

A Novel Concept of Fuzzy Subsemigroup (Ideal)

“…Then, Van de Vel [2] gave a systematic study of the theory of convexity in 1993. Rosa [3] extended the concept of convexity to the fuzzy situation in 1994, which is said to be an L-convex structure later, and the relevant results are shown in [4][5][6][7][8]. In addition, Maruyama [9] studied the concept of convexity on a completely distributive lattice in 2009.…”