the integers have additional properties which do not generalize to all commutative rings. an important example is the fundamental theorem of arithmetic, which says that every positive integer can be factored uniquely into prime numbers. unique factorizations do not always exist in other rings, but noether found a unique factorization theorem, now called the "lasker–noether theorem", for the ideals of many rings. much of noether's work lay in determining what properties "do" hold for all rings, in devising novel analogs of the old integer theorems, and in determining the minimal set of assumptions required to yield certain properties of rings.