Studies in Rings generalised Unique Factorisation Rings

-108-
M annihilated by I, I~(lV~) =
is the injective hull of M.
00
n
U annE(M)I , where E(M)
n=l
Theorem 4035 ..
Let P be a height 1 prime ideal of a GUFR. Then;'
P satisfies the right second layer condition.
P roof:
Assume that there exists a prime ideal Q of R
such that Q ( P and Q = ann M for some finitely generated
uniform right R-module M containing a copy U of a non
zero right ideal of Rip. Since P is height 1 prime, Q
is a minimal prime
ideal of R and so Q contains no
normal invertible Ldea k, Let I be the normal invertible
ideal contained in P. Put J = I+Q. Then J/Q is an
invertible ideal of R/Q and so it has the right AR
property.
Since M is an R/Q module,by the above lemma
00
we have E(M) = U
n=l
annE(M) (J/Q)n. But M is finitely
generated and is contained in E(M). This together with
the fact that [annE(M) (J/Q)n) is an ascending chain of
submodules of E(M) implies that there exists a positive
integer k such
that M S annE(M) (J/Q)k, ioe., M(J/Q)k=o
which implies (J/Q)k ~ ann M = Q and Jk/Q (Q. Hence