Publ. Math. Debrecen 75/3-4 (2009), 419–435 Jordan loop rings By BRADLEY C. DART (Newfoundland) and EDGAR G. GOODAIRE (Newfoundland) Abstract. A commutative loop or ring is said to be Jordan if it satisfies the identity (x 2 y)x = x 2 (yx). We show that the loop ring of a Jordan loop L is Jordan and not associative only if the characteristic of the coefficient ring is even and call such a loop ring Jordan (RJ, for short). While Jordan loops are in general not power associative, RJ loops are. We give various constructions of finite RJ loops and conjecture that these exist only when they have order divisible by four. We also conjecture that RJ loops are precisely those commutative loops in which squares are in the left nucleus. 1. Some history The title of this paper is inspired by another, “Alternative Loop Rings,” which appeared in this journal over twenty-five years ago [Goo83]. This was the first paper exhibiting a class of loop rings satisfying an “interesting” identity, other than associativity, and the present work is of a similar nature. In the interim, the subject of nonassociative 1 loop rings (and their underlying loops) has developed substantially. Whereas there was once reason to believe that nonassociative loop rings satisfying nonassociative identities could not exist, it turns out that virtually any identity of Bol–Moufang type is satisfied by some nonassociative loop ring [DG09]. There are RA loops, whose loop rings in all characteristics satisfy the Moufang identities (but not associativity), RA2 loops, whose loop rings have the Mathematics Subject Classification: Primary: 20N05; Secondary: 17C50, 05B15. Key words and phrases: Jordan ring, Jordan loop, loop ring, Latin square. This work was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada and completed while the first author held an Undergraduate Student Research Award from NSERC. 1 In this paper, “nonassociative” means “not associative.”