## Loops whose loop rings are alternative.(English)Zbl 0582.17015

Let R be a non-trivial commutative and associative ring with identity such that 2a$$\neq 0$$ for every $$0\neq a\in R$$. A loop L is said to be an RA-loop if the loop ring RL of L over R is alternative but not associative. In this case, L is a Moufang loop, the nucleus N(L) and the centre C(L) of L coincide, $$a^ 2\in N(L)$$ for every $$a\in L$$ and, for all a,b$$\in L$$, $$(a,b)=1$$ iff $$(a,b,c)=1$$ for any $$c\in L$$. Moreover, the associator and the commutator subloop of L are equal subgroups of order 2 of C(L) and the loop L has the property LC (i.e., if a,b$$\in L$$ and $$ab=ba$$ then $$\{a,b,ab\}\cap C(L)\neq \emptyset).$$ Conversely, a non- associative Moufang loop is an RA-loop iff it has LC and a unique non- identity commutator.
Reviewer: T.Kepka

### MSC:

 17D05 Alternative rings 20N05 Loops, quasigroups
Full Text:

### References:

 [1] Bruck R.H., A Survey of Binary Systems, Ergebnisse der Mathematik und ihrer Grenzgebiete 30 (1958) · Zbl 0081.01704 [2] Chein O., Trans. Amer. Math. Soc 188 pp 31– (1974) [3] Chein O., Arch, der Math 188 pp 121– (1973) · Zbl 0261.20063 [4] Chein O., Memoirs Amer. Math. Soc 197 (13) (1978) [5] Chein O., Comm. in Algebra 13 (13) pp 1– (1985) · Zbl 0553.17005 [6] Chein O., Proc. Amer. Math. Soc 33 (13) pp 29– (1972) [7] Goodaire E.G., Publ. Math. Debrecen 30 pp 31– (1983) [8] Hall M., The Theory of Groups (1959) · Zbl 0084.02202 [9] Paige L.J., Proc. Amer. Math. Soc 6 pp 279– (1955) [10] Schafer R.D., Introduction to Non-Associative Algebras (1966) · Zbl 0145.25601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.