Therefore, the ideals of satisfy the ascending chain condition, where ideals are ordered by set inclusion. Hence is a Noetherian ring. See also Artinian Ascending chain condition for principal ideals Krull dimension Maximal condition on congruences Noetherian Notes References Atiyah, M. F., and I. G. MacDonald, Introduction to Commutative Algebra, Perseus Books, 1969, Michiel Hazewinkel, Nadiya Gubareni, V. V. Kirichenko. Algebras, rings and modules. Kluwer Academic Publishers, 2004. John B. Fraleigh, Victor J. Katz. A first course in abstract algebra. Addison-Wesley Publishing Company. 5 ed., 1967. Nathan Jacobson. Basic Algebra I. Dover, 2009. External links Commutative algebra Order theory Wellfoundedness