ABSTRACT
R denotes a commutative ring with unity 1R. 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.
