ABSTRACT
Abstract In this chapter, we give an account of known results regarding the structure of the loop of units of an alternative loop ring.
Key words: loop, alternative loop ring, loop of units, normal complements
2000 MSC: Primary 20N05; Secondary 17D05, 16S34, 16U60
Let R be a commutative (and associative) ring with unity and let L be a loop (see Definition 27.1 below). The loop algebra of L over R was introduced in 1944 by R.H. Bruck [Bru44] as a means of obtaining a family of examples of nonassociative algebras and it is defined in a way similar to that of a group algebra; i.e., as the free R-module with basis L, with a multiplication induced distributively from the operation in L.
