A subring I of a ring R such that for every a in R and every x in I both ax and xa are in I. Where the multiplication is not commutative, it is possible that one of these conditions holds, but not the other. In these cases left ideal is used when ax always lies in I, and right ideal when xa always lies in I. If no qualification is made, the assumption is that the ideal is two-sided. The multiples of any integer form an ideal in the ring of integers.