I is finite, then the converse is true, i.e., every ideal of R is of this form. However, if I is infinite and the rings Ri are non-trivial, then the converse is false: the set of elements with all but finitely many nonzero coordinates forms an ideal which is not a direct product of ideals of the Ri. The ideal A is a prime ideal in R if all but one of the Ai are equal to Ri and the remaining Ai is a prime ideal in Ri. However, the converse is not true when I is infinite. For example, the direct sum of the Ri form an ideal not contained in any such A, but the axiom of choice gives that it is contained in some maximal ideal which is a fortiori prime. An element x in R is a unit if and only if all of its components are units, i.e., if and only if pi&hairsp;(x) is a unit in Ri for every i in I. The group of units of R is the product of the groups of units of the Ri. A product of two or more non-trivial rings always has nonzero zero divisors: if x is an element of the product whose coordinates are all zero except pi&hairsp;(x) and y is an element of the product with all coordinates zero except pj&hairsp;(y) where i ≠ j, then xy = 0 in the product ring. References Ring theory Operations on structures