On the Significance and the Extension of the Chinese Remainder Theorem

A set of elements is said to form a group under a composition rule R or simply an R-group if

(i) for every two elements a and b of the set aRb is also an element of the set, i.e. the set is closed under R;

(ii) R is associative, i.e. for any three elements a,b,c, of the set (aRb)Rc=aR(bRc);

(iii) there exists an element e, called the identity element, such that for every element a of the set ake=eRa=a;

(iv) to every element a, there exists an element x=a^-1 called the inverse of a, such that aRx=e. The group is called Abelian if R is commutative.