Noetherian and Artinian Rings

R denotes a commutative ring with unity 1 R . R is said to be noetherian if every ideal of R is finitely generated. Equivalently , Every nonempty collection of ideals of R has a maximal element . R satisfies the ascending chain condition (a. c. c) on ideals . Some properties of noetherian rings are pointed out. Hilbert’s theorem: ‘If R is noetherian, so is R [ x ] ’, is proved. Artinian rings satisfying the descending chain condition (d. c. c) on ideals are also described. It, so, happens that ℤ is noetherian but not artinian .