Bipolar complex fuzzy submodules

Abstract

In this manuscript, we interpret the theory of bipolar complex fuzzy submodule (BCFSM) of a provided classical module over a ring by utilizing the well-known notion of bipolar complex fuzzy set (BCFS). We also devise the sum of two BCFS and investigate its related proposition for studying a few fundamental properties of the initiated BCFSM. Moreover, we investigate the cartesian product of two BCFS for developing properties of BCFSM. We also investigate that if a BCFS Ӈ_1 is a BCFSM of Ӎ_1, then image Ӻ(Ӈ) is a BCFSM of Ӎ_2 and the preimage Ӻ^(-1) (Ӈ) is a BCFSM of Ӎ_1, where, Ӎ_1 and Ӎ_2 are two modules and Ӻ:Ӎ_1→Ӎ_2 is a module homomorphism.