By Ralph K. Amayo

ISBN-10: 3540105735

ISBN-13: 9783540105732

Those are the court cases of the Southern Illinois Algebra Conference.The convention was once held on the Southern Illinois college at Carbondale on April 18 and 19, 1980 as a part of the detailed Week In Algebra , April14-19, 1980. The convention used to be made attainable during the curiosity, monetary help and cooperation of the dep. of arithmetic, collage of Liberal Arts, Graduate tuition and workplace of foreign schooling at Southern Illinois college at Carbondale.

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**Additional info for Algebra Carbondale 1980 proceedings. Lie algebras, group theory, and partially ordered algebraic structures**

**Example text**

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.

### Algebra Carbondale 1980 proceedings. Lie algebras, group theory, and partially ordered algebraic structures by Ralph K. Amayo

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