Studies in Rings generalised Unique Factorisation Rings

-6-
If both conditions hold, R is said to be
Noetherian.
Example 1 0 6 .
It is easy to observe that the 2 x 2 matrices of
the form [~~
where a € Z and b,c E. Q make a ring which
) 1
is right Noethcrian but not left Noetherian.
Proposition 1 07.
Let B be a submodule of A. Then A is Noetherian if
and
only if 8 and A/S are both Noetherian.
Corollary 1.8.
Noetherian.
Any finite direct sum of Noetherian modules is
Corollary 1.9.
If R is a Noetherian ring, all finitely generated
right R-modules are Noetheriano
Definition 1010.
Given a ring R and a positive integer n, we use
M (R) to denote the ring of all n x n matrices over R.
n
The standard n x n matrix units in M (R) are the matrices
n
e .. (for i,j = 1,2, •.. ,n) such that e .. has 1 as the i_jth
1) 1J
entry and 0 elsewhere.