sentences:  The elements of the alternating group, called even permutations, are the products of even numbers of transpositions.The identity map, an empty product of no transpositions, is an even permutation since zero is even; it is the identity element of the group.The rule "even × integer = even" means that the even numbers form an ideal in the ring of integers, and the above equivalence relation can be described as equivalence modulo this ideal.