Studies in Rings generalised Unique Factorisation Rings
Studies in Rings generalised Unique Factorisation Rings
Studies in Rings generalised Unique Factorisation Rings
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Corollary 30370<br />
Let R be a semiprime GUFR with the quotient r<strong>in</strong>g Q.<br />
Also suppose that f(l) £<br />
R'for every right and left R<br />
endomorphism f of Q. Then R is classically <strong>in</strong>tegrally<br />
closed <strong>in</strong> Qo<br />
Examples 3.38<br />
( 1) Corollary 306 states that M (R) is a GUFR, whenever<br />
n<br />
R is a GUFR. Thus for any commutative Noetherian <strong>in</strong>tegral<br />
doma<strong>in</strong> RJMn(R)<br />
is a prime GUFR.<br />
(2) Let R = k[t,y] be the polynomial r<strong>in</strong>g <strong>in</strong> two<br />
commut<strong>in</strong>g <strong>in</strong>determ<strong>in</strong>ates over a field k of characteristic<br />
zero. Let be the derivation 2y ~t + (y2+t) ~y. Then R<br />
has only two -prime ideals, namely (y2+ t+l) and tR + yR.<br />
The only height 1 primes of R[x, J ] are the extensions of<br />
these two d -prime ideals o It is easy to see that these<br />
extensions conta<strong>in</strong> normal <strong>in</strong>vertible Ldea Ls and so R[x, cl ]<br />
is a GUFf{. But R[x,J] is not an NUFR [2, example 5 02.].
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