A Noetherian ring is a ring that is both left and right Noetherian. A ring is left Noetherian if every ascending sequence of left ideals eventually becomes constant, and right Noetherian if the corresponding condition holds for right ideals. These are the respective ascending chain conditions (Wisbauer 1991, p. 26).
For a ring 
with multiplicative
identity, the following are equivalent (Wisbauer 1991, p. 221).
1. 
is left Noetherian.
2. Every left ideal of 
is a finitely generated
module over 
.
3. Every nonempty set of left ideals has a maximal element with respect to inclusion.
The corresponding statements for right ideals characterize right Noetherian rings. For a commutative ring, these conditions coincide and can be stated using ideals. For a noncommutative ring, conditions on two-sided ideals alone do not give the left or right Noetherian condition.
For example, the ring of integers 
is Noetherian, since every ideal
is generated by one element. It is not an Artinian
ring, since the ideals 
form a strictly descending sequence
as 
increases.
