FTP, name this algebraic structure with two compositions, multiplication and the abelian group addition, which is also not a field.
If there exists a number n greater than one such that for every element x in this construct, x to the n equals x, then it is a Jacobson one.  If every element in it is idempotent for multiplication, it is a boolean one. The Wedderburn-Artin theorem specifies when this construct is both artinian and lacking nonzero proper ideals; in that case it is called simple.  FTP, name this algebraic structure with two compositions, multiplication and the abelian group addition, which is also not a field.
If there exists a number n greater than one such that for every element x in this construct, x to the n equals x, then it is a Jacobson one.  
{ring}