A ring (R, +, *) is said to be a non-associative ring if (R, +) is an additive abelian group, (R, *) is a non-associative semigroup (i.
But RQ will only be a non-associative ring without identy.
then Q is a Loop) that the quasigroup ring RQ will be a non-associative ring with unit.
The quasigroup ring ZQ is a non-associative ring without unit element.
The smallest non-associative ring without unit is quasigroup ring given by the following example.
Clearly the quasigroup ring ZQ is a non-associative ring.
In our view groupoid rings and quasigroup rings are the rich source of non-associative SNA-rings without unit since all other rings happen to be either associative or non-associative rings with unit.
Thirty-three papers from the July 2003 conference on non-associative algebra held in Mexico present recent results in non-associative rings
and algebras, quasigroups and loops, and their application to differential geometry and relativity.