311
Internat. J. Math. & Math. Sci.
Vol. i0 No. 2 (1987) 311-314
ON SOME PROPERTIES OF POLYNOMIAL RINGS
H. AL-EZEH
Department of Mathematics
University of Jordan
Amman
Jordan
(Received February 4, 1986 and in its revised form April 29,1986)
ABSTFCT.
For a commutative ring with unity R, it is proved that R is a PF
an(a),
and only if the annihilator,
proved that the polynlmial ring,
ring if
for each a e R is a pure ideal in R, Also it is
R[X],
is a PF-ring if and only if R is a PF-ring.
Finally, we prove that R is a PP-ring if and only if R[X] is a PP-ring.
KEY WORDS AND PHRASES.
idempotent elements.
Polynomial Rings, Pure ideal, PF-ring, PP-ring, R-flatness, and
1980 AMS SUBJECT CLASSIFICATION CODE: 13B.
INTRODUCTION.
1.
All our rings in this paper are commutative with unity.
An ideal I of a ring R is
called pure if for any x e I, there exists y e I such that xy
PF-ring if every principal ideal aR is a flat R-module.
an(a),
A ring is called a
One can easily show that aR is
if every principal ideal aR is a projective R-module.
projective if and only if the annihilator,
x
A ring R is called a PP-ring
is generated by an idempotent element,
(see
First, we state a proposition characterizing flat R-modules elementwise.
well known result in commutative ring theory, (see
PROPOSITION I.
An R-module M is a flat R-module if and only if for any pair of fi-
nite subsets {x
1, x 2,
n
Y. x.a.
1 1
i=l
n
such that E b a i
j
i=l
This is a
[3]).
Xn
and
{al,
a2,...
0
there exists elements z
0
j=l
an
of M and R respectively, such that
z
k
e M and
bij
e R; i
1,2,...,k
k
2,
k, and x i
E z b i’ i
j
j=l
n.
2
In the following theorem we establish that R is a PF-ring if and only if ann(a)
R
for each a e R is B pure ideal.
THEOREM I.
For any ring R, R is a PF-ring if and only if
an(m)
for each m e R
is a pure ideal.
PROOF.
Let x I, x 2,...,x e
n
there exists m I,
n
i=Z!
mmia i
0.
n and
al, a2,...
mn e R such that x i
Hence m e
ann(R i=Imiai )"
m2,...
mim
an e
i
n
R with
0.
i=II xiai
I, 2,...,
n.
So
Then
H. AL-EZEH
312
n
Since
n
bm.m
x
m m
2
i
i
m.
n
These elements satisfy
II=l bmlai
0 and
Therefore mR is a flat R-module
n
ann(m).
Conversely, let b
O.
Then mb
Since bR is a flat R-module, there exists
R
0 and b
c E bR and d e R such that dm
Cld
R.
bm2,..., bmn
Now take m e mR and bm I,
such that bm
ann(=imiai)_
R
is a pure ideal, there exists b e
ann(iElmfai)
R
Now c
cd.
Clb,
so b
cd
bCld.
Moreover
Therefore ann(m) is a pure ideal.
R
be a finite set of pure ideals df a rin R, then
Let I.i,
e ann(m).
R
LEMM I.
In
12
n
n
I. is a pure ideal.
J
j=1
PROOF. Let x e J. Then x e I. for each j. Thus there exists Yl
Y2 e
J
x.
J
and
Then y
xy
e
I,
x,
n.
with
2,
ylY2...yne
xyj
without nonzero nilpotent elements) ring. Let
Let R be a reduced
h(X)
h +
+
+ hn e R[X]. Than and h(X)) NIX] where N is the annihilator
0
J
II’
12
Yn In
xn
hIX
R[XJ
of the ideal generated by h 0, h
Moreover if f(X)
a
/
0
alX
+
hn,
m
.+
a X
m
ann(h0’R hI’’’’’ hn --i--0an(h i).
that is N
e ann (h(X)) then a h
i j
R[X]
n (see [4]).
I, 2
LEMMA 2. Let R be a PF-ring, then R is reduced.
PROOF. Let a be a nilpotent element in R, a
O.
n
such that a
integer greater than
0. Hence a
2
0 for all i
m
and
ann (an_
pure, there exists b e
Let n be the least positive
ann(an-l).
R
with ab
Now o
a.
ba
n-
n-l
)- is
Since ann(a
a
n-1
R
since ba
a.
R
Thus R is reduced.
Contradiction.
THEOREM 2.
The ring of polynomials,
R[X],
is a PF-ring if and only if R is a
PF-ring.
Let f(X)
PROOF.
a
Since R[X] has no nonzero
e
a
By Lemma I, J
i
i, 2,
+amXm
alX +
e ann (h(X)) where
h(X)
X
R
ho+hiX+...+ hnXn"
nil.potent elements,
i
,m
o, I, 2,
Hence there exist b I,
is pure.
m.
+
ann(h )
N
J
0
b2,..., bm
e J such that
Now our aim is to find c e J such that c f(X)
f(X).
aib i
We construct
this element inductively.
First,
a0b 0
a
Consider
0.
+
alX )( b 0 + b blb 0
a0b 0 + aob a0bob + alb0X + albl X alboblX
a +
a0b aob + alboX + alX alboX
0
a + a
iX’
0
a
Let c
b
0
+
b
0
blb0,
a
(a 0 +
0
then
alX + a2 X2
alX)C + b2(l
+
+ b2
Clb 2
Cl)(ao+al X)
+
)( c
a i,
a2ClX2 + a2b2x2 a2b2ClX2
SOME PROPERTIES OF POLYNOMIAL RINGS
a
a
+
0
c
Similarly,
c
c
m
a
0
alX + a2ClX2
alX + a2 X2
+
0
+
m-i
+
+ b2
c
2
b
Clb2,...
c
m
m-
O, I, 2,..., m.
i
c
Thus there exist c
m
R[X]
a
i
0
+
+...+
alX
Cl,...,
Cm
aiX
I
e J.
f(X).
e J with cf(X)
Let a e P and b e ann(a).
R
Since R is a PF-rng there exists
(a).
g(X)
c
Moreover c o
Conversely, assume R[X]
Then b e ann
and
ibm
...+ ai Xi)
alX +
a2cl X2
a2b2X2
+
313
is a PF-ring.
o+
ClX +
b.
Hence bc 0
c
with b g(X)
+
ckXk
e ann
b and
COa
(a)
R[X]
O.
Consequently, R is a PF-ring.
R is a PP-ring if and 8nly if R[X] is a PP-ring.
It is enough to show that ann (f(X)) is generated by an idempotent
R[X]
Since R is reduced
+a nX
+
a +
element in R] where f(X)
0
THEOREM 3.
PROOF.
n.
alX
N[X]
(f(X))
ann
where N is the annihilator of the ideal generated by
R[X3
a0, al,...
N
an.
R(aO
ann
a
al,
n
n
n_ann(a i)
R
i =0
2
i= 0
e.1
e.R,
1
ele2...en
e.I because R is a PP-ring.
)R
eR, where e
Hence ann (f(X))
R[X]
ele2...en
eR[X],
e
2
Conversely, let R[X] be
(a)L
RxJ
an
is a PP-ring
then b
2
0
Thus b
a PP-ring, let a e R, then consider ann(a).
o
+
2
g(X)R[X] where g(X)
g(X)
Let b e
anD(a)
blX +...+ bmXm
Since R[X]
R
We claim ann(a)
R
bo
b
e
boR.
)(
,
Co + ClX +’’’+ ct xt
If g(X)
then ba
)"
b
0
+
0.
Therefore b
blX
So b e
+...+
b
g(X)R[X3.
b0c O,
that is
be bRo
b0R.
boa
For the other way around, let b e
O. That is b e and (a). Thus ann(a)
K
R
Then b
b0R.
boC 0
for some c o e R.
xm
m
Since
314
H. AL-EZEH
REFERENCES
i.
EVANS, M.
2.
VASCONCELOS, W. On finitely generated flat modules, Trans. Amer. Math. Soc., 138
(1969) 505-512.
3.
LARSON, M. and MCCARTHY, P. Multiplicative Ideal Theory, Vol 43 in pure and applied
lthematics, Academic Press, New York and London (1971).
BRER, J. Power series over commutative rings, Lecture notes in pure and applied
Mathematics 64, Marcel Dekker, New York and Basel (1981).
4.
On commutative PP-rings, Pac. Jour. Math., 41(1972) 687-697.