Internat. J. Math. & Math. Sci.
(1988) 9-14
VOL. ii NO.
TWO PROPERTIES OF THE POWER SERIES RING
H. AL-EZEH
Department of Mathematics
University of Jordan
Amman, Jordan
(Received July 31, 1986 and in revised form October 29, 1986)
For a commutative ring with unity, A, it is proved that the power series ring
ABSTRACT.
A
X
is a PF-ring if and only if for any two countable subsets S and T of A such that
S Sann(T), there exists c e ann(T) such that bc
A
b for all b e
S.
Also it is proved
A
that a power series ring A
X
is a PP-ring if and only if A is a PP-ring in which
every increasing chain of idempotents in A has a supremum which is an Idempotent.
KEY WORDS AND PHRASES. Power series ring, PP-ring, PF-ring, flat, projective, annihilator ideal and idempotent element.
1980 AMS SUBJECT CLASSIFICATION CODE. 13B.
i.
INTRODUCTION.
Rings considered in this paper are all commutative with unity.
power series ring over the ring A.
principal ideal is a flat A-module.
Let A
X
be the
Recall that a ring A is called a PF-ring if every
Also a ring A is called a PP-ring if every principal
ideal is a projective A-module.
It is proved in Ai-Ezeh [I] that a ring A is a PF-ring if and only if the annihilator of each element a e
A, ann(a),
A
exists c e ann(a) such that bc
b.
is a pure ideal, that is for all b e
ann(a) there
A
A ring A is a PP-ring if and only if for each a e A,
A
In Brewer [3], semihereditary
A
power series rings over von Neumann regular rings are characterized. In this paper we
ann(a) is generated by an idempotent, see Evans [2].
characterize PF- power series rings and PP- power series rings over arbitrary rings.
For any reduced ring A (i.e. a ring with no nonzero nilpotent elements), it was
proved in Brewer et al. [4] that
ann
A
(a
X0
+
alX +
...)
N X]]
where N is the annihilator of the ideal generated by the coefficients a O,
2.
MAIN RESULTS.
al,...
H. AL-EZEH
I0
Any PF-ring A is a reduced ring.
LEMMA I.
Assume that there is a nonzero nilpotent element in A. Let n be the least
n
Because A is a PF0. So a e
such that a
positive integer greater than
A
nn-I
0
a
(ab)
a. Thus a
such that ab
ring there exists be ann
A
n-i
0.
since ba
Contradiction. So any PP-ring is a reduced ring.
PROOF.
ann(an-l).
n-lbn-I
(an-I
THEOREM 2.
The power series ring A
{bo,
countable sets S
b I, b
X
2
exists c e ann(T) such that b.c
is a PF-ring if and only if for any two
{a 0,
and T
al, ...}
ann(T), there
such that S
A
O, i, 2
b.i for i
1
A
PROOF. First, we prove that A X is a PF-ring.
Let g(X)
b +
+
and
0
b2X
+
f(X)
a
g(X)
ann
0
alX +
and let
(f(X)).
Then g(X) f(X)
0.
AX
The ring A is inparticular a PF-ring because for all b e
ann(a), there exists
A
c e
ann(a) such that bc
A
b.a.
I S
So by Lemma
b.
O,
for all i
..;
I, A
is a reduced ring.
O,
j
Thus
2,
So
{b0,
b
ann(ao,
A
).
a
So by assumption, there exists
ann(a 0, a
b for all i
such that b c
Hence g(X)c
O,
g(X)
I’
i
i
A
and c e ann (f(X)).
Consequently the ring A X is a PF-ring. Conversely, assume
c e
AEX
A [IX]] is a PF-ring.
b
Let
_c
{b0,
Then g(x) f(X)
O.
ann(a0,A
...)
al,
+ blX +
+ al+
........
Let g(X)
Therefore g(X) e ann
b
(f(X)).
and f(X)
0
a
Thus there exists h(X)
Co+ClX+...
AEX
g(X).
(f(X)) such that g(X) h(X)
in ann
0
AEx
Consequently, h(X) f(X)
c.a.
and
bic
So c
o
0
0,
for all i
for all j Z i.
0
e ann(a O, a I,
I)
0 and g(X) (h(X)
...) and
j
{Co, c
b.1 for
Hence
bic 0
0.
Since
0, i, 2
A is reduced,
anx(a 0,
all i
O, I,
0
bi(c0-1)
and
a
and
for all i
O.
bi(c0-1)
Therefore the above
condition holds.
Because any PP-ring is a PF-ring, every PP-ring is a reduced ring.
ring A, a partial order relation can be defined by a s b if ab
a
2.
On a reduced
The following
lemma is given in Brewer[3] and Brewer et al.[4].
LEMMA 3.
PROOF.
ab
a
2
defined above on a reduced ring A is a partial order.
The relation
Clearly the relation s is reflexive.
and ba
b
2
So
a-b
2
a
2
2ab
+ b2
Now assume a
O.
b and b
a.
Because A is reduced a
Then
b
O,
TWO PROPERTIES OF THE POWER SERIES RING
or a
To prove transitivity of E, assume a
b.
b and b
Ii
So ab
c.
a
2
and bc
b
2
Consider
(ac
ab)
2 2
a (c
2
2
=a (c
2cb + b
2
b
2
2
b)(c + b)
ab(c
0
b)
because b(c
Since A is reduced, ac
0.
THEOREM 4.
The power series ring
AX
0 or ac
ab
ab
a
2
Therefore a < b.
is a PP-ring if and only if A is a PP-ring
in which every increasing chain of idempotents of A with respect to N has a supremum
which is an idempotent element in A.
Assume A
PROOF.
idempotents in A
X
X
Let a e A.
is a PP-ring.
eA [[
(a)
are in A, ann
X
AX
0, rea
ea
(a).
b e ann
for all r e A.
0
ann(a).
A
Hence eA
eg(X) for some g(X)
Thus b
b
X
A
That is b e eA.
0
+
chain of idempotents in
A,
(e
ann
AEx
Since
eei
So e. N
e.1
for all i
+
(e
AEx 0
y
is a PP-ring and
eA. Because
We claim ann(a)
A
Now let b e ann(a). Hence
A
eb
Consequently, b
+
O.
blX
X
X
i
0
0e
< e
be an increasing
2
is a PP-ring and since idempotents of A
Now we claim
e
sup{e O,
e
2
e
e
i
i
}.
Let y be an upper bound of {e 0, e
O,
e
iX +
...).
for some c e A.
ec
Consequently,
ce)(l
(I
e)
y(l
e
ec
e)
+
ec
i- e
So
e
y.
Therefore
e
...}.
sup{e O, e
To prove the other way around, consider
a%xf(X))
Hence
ann
(f(X))
AEX
an(a O,
a
X
...}.
O, I,
y e ann
Thus
+
e)
e
eA
Because A
Ao
e X
ei(l
y for i
Hence
+
0
0,
1
X
Whence A is a PP-ring.
To complete the proof of this direction, let e
are in
Since A
ann(a 0
A
=/
i=O
a
ann(a i)
)X
where f(X)
a
0
+
alX +
So
12
H. AL-EZEH
e.A, e.
e.
i=0
because A is a PP-ring.
Let
do=
e
0, d
d
e0e
d
n
e
n-I n
One can easily check that
f
e.A=
i=0
d. A
i=O
Also it is clear that
o _>
d
> d
i- d
o
< I-
d
2
Therefore
_<
d
1-
d
2
By assumption, this increasing chain of idempotents has a supremum which is an idempotent.
Let
Sup{l
(I
d
di)
o
d I,
d
d
d 2, ...}
d.
0,
fo,:all i
i
We claim that
Now
d > d
d)d.
So (i
i.
(I -d)A
Thus
(I
d)A
i=0
diA.
Hence
d.
diA
for all i
0,
diA-
i=O
Let y
d)A.
(i
diA
Then y
i,0,1,
diY i,
Consequently
(i
di)(l
+
y)
diY
l-d
Because
Therefore
yd
i
d
d
2
2. diYi
diYi
1
i
y
y"
for all i
0,
i
d
i
y
So
TWO PROPERTIES OF THE POWER SERIES RING
Because
Sup{l
d
$
d
Hence
That is
0.
dy
y
g
(I
d
},
So
d I,
d(l
y)
d
Thus y(l
d)
y
yd
d
y.
d)A.
O
2
dy
y
d.A
Therefore
13
(i
d)A.
i;O
Consequently,
(f(X))
A[[X]]
ann
Therefore
A[[X]]
(I- d) A[[X]]
is a PP-ring.
REFERENCES
i.
AL-EZEH, H.
On Some properties of Polynomial rings.
I.J.M.M.S
To appear
Pac. J. Math. 41(1972) 687-697.
2.
EVANS, M.
3.
BREWER, J.
4.
RUTTER, E. and WATKINS, J. Coherence and weak global dimension of
BREWER, J.
R [[X]] when R is Von Neumann regular, J. of Algebra 46(1977) 278-289.
On commutative PP-rings.
series over commutative rings". Lecture Notes in pure and applied
Mathematics No. 64, Marcel Dekker, New York and Basel (1981).
"Power