Studies in Rings generalised Unique Factorisation Rings

Chapter-3
EXTENSIONS AND RINGS WITH MANY NORMAL ELEMENTS ,
INTRODUCT ION
In this chapter, we discuss rings which are extensions
of GUFRs
namely the finite centralising extensions, Ore
extension and the ring of polynomials twisted by a derivation.
Also we introduce the concept of rin~s with many normal
elements.
We
show that any finite centralising e xt.e n s i o n of a
GUFR is a GUFR 0 As a corollary of this result, M (R), the
n
n x n matrix ring, over a GUFR R is a GUFR. A sufficient
condition for the Ore extension, over a Noetherian ring
with Artinian quotient ring, to be a GUFR
is obtained. The
Noetherian rings with Artinian quotient rings such that
the Ore extensions over them are prime GUFRs a r e characterised.
The skew polynomial rings over some special Noetherian rings
are investigatedo
We
extend the concept of rings with few zero divisors[13]
in the commutative case to rings with many normal
elements
in the non-ecomrnu t a t.Ive case. By introducing the concept of
weakly invertible elements, we study some properties of
Noetherian rings with many normal elements. Also, we prove
some
results.on the integral closure of Noetherian rings, in
this chapter.