3 0 Ju l 2 00 5 STRONGLY RIGHT ALTERNATIVE RINGS AND BOL LOOPS

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. A magma (L, ·) is a set L with a binary operation · : L × L → L. A magma (L, ·) is a loop if there is a neutral element 1 ∈ L satisfying 1x = x1 = x, and if all right translations R(a) : x → xa and all left translations L(a) : x → ax are bijections. If a magma L satisfies the right Bol identity (xy · z)y = x(yz · y), then we will say that L is a Bol magma. The Bol identity can be equivalently expressed in terms of right translations: R(y)R(z)R(y) = R(yz · y). A Moufang loop is a Bol loop which also satisfies the flexible law x · yx = xy · x. For background in loops, the reader is referred to the standard references [1] and [9], and further details on Bol loops can be found in [10]. A (nonassociative) ring (R, +, ·) is right alternative if it satisfies the right alternative law xy · y = xy 2. A right alternative ring R is strongly right alternative if (R, ·) is a Bol magma. A right alternative ring of characteristic not 2 is strongly right alternative ([11], §16.1), but this is not true in general [3] [7]. An alternative ring is a right alternative ring which also satisfies the left alternative law x · xy = x 2 y. An alternative ring is strongly right alternative ([11], §2.3), but there are strongly right alternative rings which are not alternative. In a ring R with unity, a unit is an element with a (not necessarily unique) two-sided inverse. Let U(R) denote the set of all units in R. It is well-known that if R is an associative ring with unity, then U(R) is a group. More generally, if R is an alternative ring with unity, then U(R) is a Moufang loop ([5], §11.5.3). In a strongly right alternative ring with unity, it is known that if U(R) is closed under multiplication, then U(R) is a Bol loop ([4], Lem. 3.1, p. 356). Goodaire posed the following problem …