In this paper we study the two-sided ideals of the enveloping algebra U s ลฝ ลฝ .. U sl K over an arbitrary field K of characteristic zero. Starting with two basic 2 ideas, that an irreducible Lie module is generated by its highest weight vector and that the Lie module structure of U comes from its ring multiplication, we have found a ''good'' subset of U consisting of highest weight vectors for irreducible U-submodules of U so that each two-sided ideal of U is uniquely generated by at most two elements of that set. Actually, each ideal is generated as a two-sided ideal by just one element. By uniqueness, all the information about the ideal is encoded ลฝ . in the formula for its generator s . For example, we can list and classify all the prime ideals by height, determine the intersection of an ideal with the center, find the radical ideals and the radical of an ideal, and determine when two ideals are included one in another. An interesting property is that each ideal of U can be uniquely written as a product of primes. We also obtain the ''least common multiple'' and the ''greatest common divisor'' formulas for the prime ideal factorizations of the intersection and the sum of two ideals. This paper contains many other results of this nature.