[Paper Review] Strongly right alternative rings and Bol loops

Michael Kinyon, J. D. Phillips|ArXiv.org|Jul 30, 2005

Mathematics and Applications5 references3 citations

TL;DR

This paper proves that in a finite strongly right alternative ring, the set of units under multiplication and the set of quasiregular elements under circle multiplication both form Bol loops. The key result establishes that finiteness ensures closure of the unit and quasiregular sets under their respective operations, generalizing known results for alternative rings and confirming partial answers to open problems posed by Goodaire on the loop structure of these sets in nonassociative rings.

ABSTRACT

We partially answer two questions of Goodaire by showing that in a finite, strongly right alternative ring, the set of units (if the ring is with unity) is a Bol loop under ring multiplication, and the set of quasiregular elements is a Bol loop under "circle" multiplication.

Motivation & Objective

• To resolve two open problems posed by Goodaire concerning the loop structure of units and quasiregular elements in strongly right alternative rings.
• To establish that in finite strongly right alternative rings, the set of units under multiplication forms a Bol loop.
• To show that the set of quasiregular elements under circle multiplication also forms a Bol loop in such rings.
• To generalize known results for alternative rings by proving that the unit and quasiregular sets form Moufang loops in alternative rings via a new approach.
• To investigate the closure of the set of invertible elements in Bol magmas with a neutral element under finiteness assumptions.

Proposed method

• Prove that in a finite Bol magma with a neutral element, the set of elements with two-sided inverses is closed under multiplication and forms a loop.
• Use the right Bol identity and properties of right translations to show that inverses are unique and that right and left translations are bijections on the invertible set.
• Apply the finiteness condition to show that elements with right inverses must also have left inverses, ensuring closure of the invertible set.
• Leverage the flexible identity in the case of flexible Bol magmas to show that the invertible set forms a Moufang loop.
• Use the circle operation $ x \circ y = x + y + x \cdot y $ to define quasiregular elements and analyze the magma structure under this operation.
• Apply the same closure and loop structure arguments to the quasiregular set under the circle operation, showing it forms a Bol loop in finite strongly right alternative rings.

Experimental results

Research questions

• RQ1 Does the set of units in a finite strongly right alternative ring with unity form a Bol loop under multiplication?
• RQ2 Does the set of quasiregular elements in a finite strongly right alternative ring form a Bol loop under the circle operation?
• RQ3 Can the closure of the unit set in a Bol magma with a neutral element be guaranteed under finiteness?
• RQ4 Does the invertible set in a flexible Bol magma with a neutral element form a Moufang loop?
• RQ5 Can the results for strongly right alternative rings be extended to infinite rings, or are finiteness assumptions essential?

Key findings

• In a finite strongly right alternative ring with unity, the set of units $ \mathcal{U}(R) $ forms a Bol loop under ring multiplication.
• In a finite strongly right alternative ring, the set of quasiregular elements $ \mathcal{Q}(R) $ forms a Bol loop under the circle operation $ x \circ y = x + y + x \cdot y $.
• For a finite Bol magma with a neutral element, the set of invertible elements $ \mathcal{J}(L) $ is closed under multiplication and forms a loop.
• In a flexible Bol magma with a neutral element, the set of invertible elements $ \mathcal{J}(L) $ forms a Moufang loop.
• The results provide a new proof that in an alternative ring, the units and quasiregular elements form Moufang loops, using the structure of Bol magmas.
• The paper shows that finiteness is sufficient to ensure closure of the invertible set in a Bol magma, even without assuming associativity or flexibility.