given ring forms a complete modular lattice. The lattice is not, in general, a distributive lattice. The three operations of intersection, sum (or join), and product make the set of ideals of a commutative ring into a quantale. If are ideals of a commutative ring R, then in the following two cases (at least) is generated by elements that form a regular sequence modulo . (More generally, the difference between a product and an intersection of ideals is measured by the Tor functor: ) An integral domain is called a Dedekind domain if for each pair of ideals , there is an ideal such that . It can then be shown that every nonzero ideal of a Dedekind domain can be uniquely written as a product of maximal ideals, a generalization of the fundamental theorem of arithmetic. Examples of ideal operations In we have since is the set of integers which are divisible by both and . Let and let . Then, and while In the first computation, we see the general pattern for taking the sum of two finitely generated ideals, it is the ideal generated by the union of their generators. In the last three we observe that products and intersections agree whenever the two ideals intersect in the zero ideal. These computations can be checked using Macaulay2. Radical of a ring Ideals appear naturally in the study of modules, especially in the form of a radical. For simplicity, we work with commutative rings but, with some changes, the results are also true for non-commutative rings. Let R be a commutative ring. By definition, a primitive ideal of R is the annihilator of a (nonzero) simple R-module. The Jacobson radical of R is the intersection of all primitive ideals. Equivalently, Indeed, if is a simple module and x is a nonzero element in M, then and , meaning is a maximal ideal. Conversely, if is a maximal ideal, then is the annihilator of the simple R-module . There is also another characterization (the proof is not hard): For a not-necessarily-commutative ring, it is a general fact that is a unit element if and only if is (see the link) and so this last characterization shows that the radical can be defined both in terms of left and right primitive ideals. The following simple but important fact (Nakayama's lemma) is built-in to the definition of a Jacobson radical: if