Abstract
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.