Krull's theorem on these states that in a commutative unit ring, the height of every proper one generated by n elements is at most
In order theory, this is a subset of a partially ordered set that is both lower and directed, and the dual notion is called a filter. Krull's theorem on these states that in a commutative unit ring, the height of every proper one generated by n elements is at most n. They were first proposed by Dedekind in Lectures on Number Theory to generalize the number of this type introduced by Kummer. The most general form of the Chinese Remainder Theorem concerns these when they are pairwise coprime, and they are the object in ring theory analogous to normal subgroups. In the ring of polynomials all of these are principal, which is to say generated by one element. FTP, name this mathematical object, defined in algebra as an additive subgroup of a ring that is stable under multiplication.
In order theory, this is a subset of a partially ordered set that is both lower and directed, and the dual notion is called a filter.
ideal