at least one of a and bn is in I for some natural number n. Every prime ideal is primary, but not conversely. A semiprime primary ideal is prime. Principal ideal: An ideal generated by one element. Finitely generated ideal: This type of ideal is finitely generated as a module. Primitive ideal: A left primitive ideal is the annihilator of a simple left module. Irreducible ideal: An ideal is said to be irreducible if it cannot be written as an intersection of ideals which properly contain it. Comaximal ideals: Two ideals are said to be comaximal if for some and . Regular ideal: This term has multiple uses. See the article for a list. Nil ideal: An ideal is a nil ideal if each of its elements is nilpotent. Nilpotent ideal: Some power of it is zero. Parameter ideal: an ideal generated by a system of parameters. Two other important terms using "ideal" are not always ideals of their ring. See their respective articles for details: Fractional ideal: This is usually defined when R is a commutative domain with quotient field K. Despite their names, fractional ideals are R submodules of K with a special property. If the fractional ideal is contained entirely in R, then it is truly an ideal of R. Invertible ideal: Usually an invertible ideal A is defined as a fractional ideal for which there is another fractional ideal B such that . Some authors may also apply "invertible ideal" to ordinary ring ideals A and B with in rings other than domains. Ideal operations The sum and product of ideals are defined as follows. For and , left (resp. right) ideals of a ring R, their sum is , which is a left (resp. right) ideal, and, if are two-sided, i.e. the product is the ideal generated by all products of the form ab with a in and b in . Note is the smallest left (resp. right) ideal containing both and (or the union ), while the product is contained in the intersection of and . The distributive law holds for two-sided ideals , , . If a product is replaced by an intersection, a partial distributive law holds: where the equality holds if contains or . Remark: The sum and the intersection of ideals is again an ideal; with these two operations as join and meet, the set of all ideals of a