Theorem 3 Let A be a -fuzzy left ideal, and let B be a fuzzy subset of R. Then A ⊙ B is a -fuzzy left ideal of R. Proof For all z 1, z 2 ∈ R, we have.
Theorem 4 Let A be a fuzzy subset, and let B be a -fuzzy right ideal of R. Then A ⊙ B is a -fuzzy right ideal of R. The following theorem is an immediate consequence of Theorem 3 and Theorem 4.
Theorem 13 Let A be a fuzzy subset of R. If for all t ∈ ( λ, μ ], A t is a subring (ideal) of R or A t = ∅, then A is a -fuzzy subring (fuzzy ideal) of R. Proof The proof can be obtained from Theorem 6.
So μ is a fuzzy right ideal of R. Since Im f ⊆ Im μ ∪ { 0 } and μ is finite valued, f is finite valued.
Theorem 20 Let A be a -fuzzy ideal of R such that A μ ≠ ∅, and let B be a -fuzzy semiprime (fuzzy primary, fuzzy semiprimary) ideal of A μ. Then A ∩ B is a -fuzzy semiprime (fuzzy primary, fuzzy semiprimary) ideal of A μ.
A is said to be a -fuzzy ideal of R if it is both a -fuzzy left ideal and a -fuzzy right ideal of R. According to the above definitions, a -fuzzy left ideal or a -fuzzy right ideal of R must be a -fuzzy subring.
Theorem 19 Let A be a -fuzzy ideal of R such that A μ ≠ ∅, and let B be a -fuzzy prime ideal of A μ. Then A ∩ B is a -fuzzy prime ideal of A μ. Proof From Theorem 1 and Theorem 6, A μ is a subring of R and A ∩ B is a -fuzzy ideal of A μ.
Theorem 5 Let A be a -fuzzy left ideal, and let B be a -fuzzy right ideal of R. Then A ⊙ B is a -fuzzy ideal of R. One of the most common methods of studying a fuzzy subring and a fuzzy ideal is by using their cut sets.
Let A be a -fuzzy prime ideal of R. Then A is a -fuzzy ideal of R. So, A t is an ideal of R or A t = ∅ from Theorem 6.
Then f − 1 ( B ) is a -fuzzy prime ideal of R. Proof From Theorem 9, f − 1 ( B ) is a -fuzzy ideal of R. Let x, y ∈ R and t ∈ ( 0, 1 ]. If ( x y ) t ∈ f − 1 ( B ), then ( f ( x ) f ( y ) ) t ∈ B. Considering B is a -fuzzy prime ideal of R ′, we have f ( x ) ∈ B t or f ( y ) ∈ B t.
