Definition 2

A ring A is said to be left Artinian (resp. left Noetherian) if the left A-module $A_s$ is Artinian (resp. Noetherian). Likewise, a ring A is said to be right Artinian (resp. right Noetherian) if the right A-module $A_d$ is Artinian (resp. Noetherian).

A ring A is right Artinian (resp. right Noetherian) if and only if its opposite ring $A^o$ is left Artinian (resp. left Noetherian). For a commutative ring A, the properties of being left Artinian and of being right Artinian coincide, and when they are hold, we say that the ring A is Artinian; we adopt an analogous convention for “Noetherian.” There exist noncommutative rings that are left Artinian but not right Artinian, and noncommutative rings that are left Noetherian but not right Noetherian (VIII, p. 14, Exercise 3).

Let A be a ring. By definition, the following properties are equivalent:

(i) The ring A is left Artinian.

(ii) Every nonempty set of left ideals of A, ordered by inclusion, has a minimal element.

(iii) Every decreasing sequence of left ideals of A is stationary.

Because of Proposition 2 of VIII, p. 3, the following properties are equivalent:

(i) The ring A is left Noetherian.

(ii) Every nonempty set of left ideals of A, ordered by inclusion, has a maximal element.

(iii) Every increasing sequence of left ideals of A is stationary.

(iv) Every left ideal of A is generated by a finite subset of A.