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By Garrett P.

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29 Proof: The proof of the following theorem is left for the reader as an exercise. 37: Let µ and λ be fuzzy normal subgroups of a group G and t ∈ [0, 1]. Let A and B be non-empty subsets of the group G. Then µ t ( A ) ⊆ A ⊆ µˆ t ( A ) i. ii. iii. µˆ t ( A ∪ B ) = µˆ t ( A ) ∪ µˆ t ( B ) A ∩ B ≠ φ ⇒ µt ( A ∩ B ) = µt ( A ) ∩ µt ( B ) iv. A ⊆ B ⇒ µt ( A ) µ t ( B ) v. vi. A ⊆ B ⇒ µˆ t ( A ) ∪ µˆ t ( B ) µ t ( A ∪ B ) ⊇ µt ( A ) ∪ µt ( B ) vii. A ∩ B ≠ φ , µˆ t ( A ∩ B ) ⊂ µˆ t ( A ) ∩ µˆ t ( B ) µ ⊂ λ ⇒ µˆ ( A ) ⊆ λˆ ( A ) viii.

If the ring Rµ has no non-zero nilpotent elements, then the fuzzy ideal µ is fuzzy semiprime. THEOREM [99]: A ring R is regular if and only if every fuzzy ideal of R is idempotent. Proof: Refer [99]. 24: A ring R is regular if and only if every fuzzy ideal of R is fuzzy semiprime. Proof: Straightforward hence left for the reader to prove. Now we recall some results on fuzzy subrings from [40]. Now for the time being we assume that R is a commutative ring with unit and M will denote a maximal ideal of R.

If Q' : R' → L is a primary fuzzy ideal of R' then f –1(Q') is a primary fuzzy ideal of R . iii. Let f : R → R' be an epimorphism of rings and Q: R → L and Q' : R' → L be a fuzzy ideals. (a) Q' is primary if and only if f –1(Q') is primary. (b) If Q is constant on ker f, then Q is primary if and only if f (Q) is primary. Now we proceed on to define the notion of weak primary fuzzy ideals. 12: A fuzzy ideal J: R → L is said to be weak primary or in short wprimary if J(xy) = J(x) or J(xy) ≤ J(yn) for some integer n > 0.

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Bernstein's analytic continuation of complex powers (1995)(en)(9s) by Garrett P.


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