Studies in Rings generalised Unique Factorisation Rings

-70-
regular element of R is regular in M (R).
n
Now the
corollary follows from theorem 3 05.
TWISTED POLYNOMIALS
In this section we study some aspects of the
~elationship between a ring R, where R is a Noetherian
ring an automorphism a,
R[x,a] = S.
and the Ore extension ring
The elements of 5 are polynomials in x
with coefficients from R written on the left of x.
We define xr = a(r)x f o r all r € R. A typical element
of S has the form, f(x) = a +alx+ •.• + a x n = a +
ono
+ ••• + x n a:-1(a), where n> 0 and a.~ R.
n - 1
The automorphism a on R can be extended to 5 by setting
a ( x) = x s 0
tha t
Definition 3.7
An a-ideal I of a ring R with an automorphism a
is any ideal I of R with a(I) ~ I. An a-prime ideal of R
is an a-ideal P such that if X and Y are two a-ideals
with Y:i S P, then either X s P or'! s P. R is said to be
an a-prime ring, if 0 is an a-prime ideal.