Prime ideals, as opposed to prime elements, provide a way to circumvent this problem. A prime ideal is a proper (i.e., strictly contained in formula_1) ideal formula_129 such that, whenever the product formula_130 of any two ring elements formula_14 and formula_15 is in formula_129, at least one of the two elements is already in formula_129. (The opposite conclusion holds for any ideal, by definition.) Thus, if a prime ideal is principal, it is equivalently generated by a prime element. However, in rings such as formula_126, prime ideals need not be principal. This limits the usage of prime elements in ring theory. A cornerstone of algebraic number theory is, however, the fact that in any Dedekind ring (which includes formula_126 and more generally the ring of integers in a number field) any ideal (such as the one generated by 6) decomposes uniquely as a product of prime ideals.