Internat. J. Math. & Math. Sci.
VOL. 12 NO. 4 (1989) 725- 728
725
EXCHANGE PF-RINGS AND ALMOST PP-RINGS
H. AL-EZEH
Department of Mathematics
University of Jordan
Amman, Jordan
(Received June I, 1988 and in revised form August II, 1988)
ABSTRACT.
Let R be a commutative ring with unity.
In this paper, we prove that R is
Let X be a completely
an almost PP-PM-rlng if and only if R is an exchange PF-rlng.
regular Hausdorff space, and let
BX
be the Stone
ch
compactlfication of X.
Then we
prove that the ring C(X) of all continuous real valued functions on X is an almost PPring if and only if X is an F-space that has an open basis of clopen sets.
we deduce that the ring C(X)
i.e. for each f
KEY
WORDS
AND
is an almost PP-ring if and only if
&C(X), there exists a unit uC(X) such that f
PHRASES.
PP-rlng,
PF-rlng,
PM-rlng,
almost
Finally,
C(X) is a U-rlng,
ulf I.
P-rlng,
pure
ideal,
exchange ring, idempotents, Stone-Cech compactlflcatlon, Boolean space and the ring of
all continuous real valued functions over a space X, C(X).
1980 AMS SUBJECT CLASSIFICATION CODES.
I.
Primary 13C13, Secondary 54C40.
INTRODUCTION.
All rings considered in this paper are commutative with unity.
Recall that R is
called a PF-ring if every principal ideal aR is a flat R-module, and it is called a
PP-rlng if every principal ideal mR is a projective R-module.
is called pure if for each x
& I, there exists
that R is a PF-rlng if and only if for a
R,
y
I such that xy
An ideal I of a ring R
It is well-known
x.
annihilator ideal, ann(a)
is pure, see
R
Also it is well-known that R is a PP-rlng if for each aR,ann(a)
R
is generated by an idempotent.
In an earlier paper we introduced almost PP-rlngs as a
ill.
Ai-Ezeh
A ring R is called an almost PP-rlng if for each aR,
generalization of PP-rlngs.
ann(a) is generated by idempotents of R.
R
almost PP-rlng if and only if for each a
e in
ann(a) such that be
In fact, one can easily show that R is an
R and b ann(a), there exists an idempotent
R
b.
R
A ring R is called an exchange ring if every element in R can be written as the
sum of a unit and an idempotent.
for example Monk
[2]
Exchange rings have been studied extensively, see
and Johnstone
[3].
Our aim in this paper is to study the
H. AL-EZEH
726
relationship between exchange PF-rlngs and almost PP-rlngs.
A ring R is called a PM-rlng if every proper prime ideal
need two more definitions.
of R is contained in a unique maximal ideal of R.
all continuous
C(X),
it is well-known that the ring of
space X,
functions over a completely regular Ilausdorff
real valued
is a PM-rlng,
To carry out our study we
see Gillman and Jerlson [4].
A compact Hausdorff and totally
disconnected space is called a Boolean (or Stone) space.
2. MAIN RESULTS.
First, we state a theorem that was proved by Johnstone [3].
A ring R is an exchange ring if and only if it is a PM-ring and the
THEOREM 2.1
space of maximal ideals of R, Max(R), is a Boolean space.
PF-rlng, there exists
c=e+u,
c ann(a)
such that bc
where e =e and u is a unit in R.
e
+ u.
b(l
e)
Hence cu
-I
eu
-I
+ I, and so
O. But bc
e)
O, a(l
e). Since ac
-I
Therefore bu
b. Consequently,
(I
e)
b, so b(l
(I
cu
e
Since R is a
R
Because R is an exchange ring,
b.
R
2
(I
bcu
e)
bu
(I
e)
PM-rlng.
Let a R, and let bann(a).
Let R be an exchange PF-rlng.
PROOF.
Then it is an almost PP.
Let R be an exchange PF-rlng.
THEOREM 2.2
b.
Since
e
e)
ub since c
ann(a), R is an almost
R
PP-rlng.
By Theorem l, R is a PM-rlng.
Hence R is an almost PP-PM-rlng.
Now we want to establish the converse of theorem 2.2.
Clearly, every almost PP-
So, by theorem 2.1, it is enough to show that the space of maximal
R, Max(R), is a Boolean space. De Marco and Orsattl [5] proved that if R is
So it is left to show that for
a PM-rlng, then Max(R) is a compact Hausdorff space.
That is for any two distinct
an almost PP-PM-rlng R, Max(R) is totally separated.
maximal ideals M and M there exists a clopen set in Max(R) containing M but not M
I.
THEOREM 2.3 Let R be an almost PP-PM-rlng. Then R is an exchange PF-rlng.
PROOF. By the above arguement, R is a PF-PM-rlng. reover, Max(R) is a compact
ring is a PF-rlng.
ideals of
Hausdorff
a
space.
Let
,
M
2Max(R)
and M
M
2.
Since
b M2 such that ab
O, see Contessa [6].
ring, there exists an idempotent e
ann(b) such that ea--a.
exist
and
M
eM2.
and
R
Since e is an idempotent,
U=D(e)-
R is a PM-rlng, there
Because R is an almost PPTherefore
{ME Max(R): eM}
eM
is a clopen set in
Max(R) containing M but not M2. So, by theorem 2.1, R is an exchange PF-rlng.
For a completely regular Hausdorff space X, the ring of all continuous real
Horeover,
valued functions, C(X), is a PM-rlng, see Gillman and Jerlson [4].
Max(C(X)), is homeomorphlc to X, the St one-ech compatlflcatlon of X. Therefore C(X)
is an almost PP-rlng if and only if R is an exchange PF-rlng.
an almost PP-rlng if and only if it is a PF-ring and
et al
[7], proved that C(X)
X
Consequently, C(X) is
is a Boolean space.
is a PF-rlng if and only if X is an F-space, where X is
called an F-space if every finitely generated ideal is principal.
that
l-Ezeh
It is well-known
X is an F-space if and only if any to nonempty disjoint cozero sets are
EXCHANGE PF-RINGS AND ALMOST PP-RINGS
Therefore, the ring C(X) is an almost PP-ring if and only if X
completely separated.
F-space and
is an
727
X is a Boolean space.
if X has an open basis of clopen sets.
In fact, BX
[s a
Boolean space if and only
Thus the ring C(X) is an almost PP-ring if and
only if X is an F-space that has an open basis of clopen sets.
Finally,
every f
C(X),
there exists a unit u
proved that the ring C(X)
THEOREM 2.4
end
is
C(X)
such that f
u
lf I.
to be a U-ring if for
In the same paper they
U-ring [f and only if X is an F-space and BX is a
a
So we get the following theorem.
Boolean space.
We
[8] defined the ring C(X)
Gillman and Ienriksen
The ring C(X) is an almost PP-ring if and only if it is a U-ring.
this
paper
by
some
giving
examples
illustrating
the
relationships
discussed above.
EXAMPLES.
I)
its
Let N be the set of positive integers with the discrete topology.
The space BN\N is a compact F-space,
Stone-Cech compactification.
Let
N be
see Gillman
Moreover, N\N is totally disconnected. Hence, the space NN
So the ring C(N\N) is an almost PP-rlng. llowever, it is not a PP-ring
’because the space \N is not basically disconnected, see Brookshear [9].
+
2) Let R be set of nonnegative reals endowed with the usual topology.
The
+
space R+\R is a compact, connected F-space, see Gillman and Henriksen [8].
Thus,
and Jerisen [4].
is Boolean.
the ring C(X) has no nontrivial idempotents.
would be an integral
divisors.
domain which
Consequently,
C(BR+\R+)
is
not
So, if it were an almost PP-ring, it
the case because
it
has
plenty of zero
is a PF-rings that is not an almost PP-ring.
3) The ring of integers is an almost PP-ring that is not a PM-ring, and so not an
exchange ring.
REFERENCES
I.
2.
3.
Some properties of polynomial rlngs. Internat. J. Math. and Math.
AL-EZEH, H.
Sci. 10(1987) 311-314.
A characterization of exchange rings. Proc. Amer. Math. SOc.
MONK, G.
35(1972) 349-353.
"Stone Spaces", Cambrid.e Studies in Advanced Mathematics No. 3,
JOHNSTONE, P.
Cambridge University Press, Cambridge (1982).
7.
GILLMAN, L. and JERISON, M.
"Rings of continuous Functions" Graduate Texts
Math. Vol. 43, Springer-Verlag, Berlin (1976).
DE MARCO, G. and ORSATTI, A.
Commutative rings in which every prime ideal
contained in a maximal ideal. Proc. Amer. Math. SOc. 30(1979) 459-466.
CONTESSA, M. On PM-rings. Communications in algebra, 10(1982) 93-108.
Some properties of the ring of
AL-EZEH, H., NATSHEH, M. and HUSSEIN, D.
8.
GILLMAN,
4.
5.
6.
in
is
continuous functions. Arch der Math to appear (1988).
L.
and HENRIKSEN,
M.
Rings
of
continuous
functions
in
which
every
finitely generated ideal is principal. Trans. Amer. Math. Soc. 82(1956),
366-391.
9.
BROOKsHEAR, J.
Projective ideals in the ring of continuous functions,
of J. Math. 71(1977) 313-333.
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