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55 PROBLEMS: 1. 2. 3. 4. 5. 6. 7. Is ZL5(3) ≅ ZL5 (2)? Can ZL5(3) be S-Isomorphic with Z7 (3)? Justify your answer Can we say QLn (m) is S-isomorphic with Z2Ln(m)? Justify your claim. Find all S-units in Z5 L9 (8). Can Z5 L9 (8) have central units? Verify Zassenhaus conjecture for the loop ring ZL11 (3). Will ZL11 (3) satisfy S-Zassenhaus conjecture? Justify. 56 Chapter 3 GROUPOID RINGS AND SMARANDACHE GROUPOID RINGS This chapter has 5 sections. In the first section we introduce the concept of groupoids and Smarandache groupoids.

27: If the loop ring RL is S-strongly right commutative then RL is Sright commutative. Proof: Direct from the very definition. We proceed on to define inner commutative loops and its Smarandache analogue. 15: Let RL be the loop ring of a loop L over R. We say the loop ring RL is inner commutative if every subring of RL is commutative. 28: Let R be a field or a commutative ring with 1. Lp(m) ∈ Lp be a loop where p is an odd prime. The loop ring RLp(m) is inner commutative. Proof: Follows from the fact Lp(m) has only subloops, which are subgroups of order 2.

Do they form a supermodular lattice? Give an example of a loop whose S-ideal forms a strongly modular lattice. Give an example of a loop ring where only S-ideals form a modular lattice but ideals do not form a modular lattice. 5 General properties of Smarandache loop rings and loop rings The determination of group of units in a group ring is an interesting one to several researchers. G. Goodaire [24 to 29] has studied this problem for loop rings in 1986. The isomorphism problem for group rings was proposed in [32].

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