### Fuzzy coprimary submodules

#### Abstract

Let

R be a commutative ring with non-zero identity and M a non-zero

unitary

R-module. We introduce the concept of fuzzy coprimary submodule as a dual

notion of fuzzy primary one and study some of their properties. A kind of uniqueness

theorem will be proved and the set of fuzzy Attached primes of a non-zero fuzzy

representable submodule of

M, the counterpart of the set of fuzzy associated primes

of a fuzzy decomposable module, will be defined. Then we deduce that for a nonzero

fuzzy submodule

μ, the (fuzzy) isolated coprimary components of μ in any fuzzy

minimal coprimary representation are independent of the choice of any minimal fuzzy

coprimary representation. Also it will be shown that, whenever

R is Noetherian, a

fuzzy prime ideal

is attached to μ if and only if is the annihilator of a fuzzy

quotient of

μ. The behavior of fuzzy attached primes with fuzzy quotient and fuzzy

localization techniques will be studied.

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