Studies in Rings generalised Unique Factorisation Rings

-13~
result.
In Noetherian rings we have the following important
Proposition 1.26.
In a
right Noetherian ring, the prime radical is
nilpotent and contains all the nilpotent right or left
ideals.
Definition 1.27.
R-module A.
Let R be a ring and S be a subset of a right
The .annihilator of S is defined as
{r E:. R I s r = 0 for a 11 s ~ s} 0
r(S), the right annihilator of S
IfS i sasubset 0 f R,
is defined as
[r E: R I s r :;: 0 for a 11 s ~ S} a nd 1eft anni h i 1 a tor 1 (s )
is defined as {r f. R I rs = 0 for all s E s} 0 A module A
is said to be faithful if annihilator of A = O.
Definition 1.28.
An R-module A
is said to be simple if A has no
proper subrnodul e s . A ring R is said to be simple if it
has no
proper ideals.