Download on projective ideals in commutative rings - Al

ICITNS 2003 International Conference on Information Technology and Natural Sciences
ON FLAT IDEALS IN COMMUTATIVE RINGS
Sarab A. Al-Taha
Department of mathematics, Faculty of Sciences
Al-Zaytoonah University, Amman Jordan
1. Abstract:
This Paper deals with the problem of flat ideals when the Ring is commutative
with identity. In this paper we show that the principal ideal <a> is flat in R if, and
only if <a> is flat in R [x] if, and only if <a> is flat in R [x1, x2] if, and only <a> is flat
R [x1, x2…xn]. Two statements one of them which concerns the principal flat ideals
in R [x] have been established.
2.Introduction:
In [1] the author proved the following result: Let R be a commutative ring with
identity, let a be a non-zero element in R, the principal ideal <a> is flat in R if, and
only if the following holds:
If a b = 0 there exists a`є R such that:
a = aa`,
(1)
a`b = 0 .
(2)
and
Through out this paper R [x] represents the polynomial ring in the indeterminate x
over a ring R and R [x, y] = R [x] [y] is the polynomial ring in the indeterminates x
and y over a ring R see [3,p293]
3. Main Results
In this section we state down the following main results.
Theorem (3.1)
Let R be a commutative ring with identity, let <a> be an ideal in R, then <a> is
flat in R if, and only if <a> is flat in R [x].
Theorem (3 .2)
Let R be a commutative ring with identity, let <a> be an ideal in R, then <a> is
flat in R if, and only if <a> is flat in R [x, y].
ICITNS 2003 International Conference on Information Technology and Natural Sciences
Theorem (3.3)
Let R be a commutative ring with identity, let <a> be an ideal in R, then <a> is
flat in R if, and only if <a> is flat in R [x1, x2…,xn].
Corollary (3.1)
Let R be a commutative ring with identity, let a є R then the following are
equivalent
(I)
(II)
(III)
(IV)
<a> is flat in R
<a> is flat in R [x]
<a> is flat in R [x1 , x 2 ]
<a> is flat in R [x1, x2…,xn].
Theorem (3.4)
Let R be a commutative ring with identity then the following are equivalent
(I)
(II)
<a> is flat in R [x]
If {ai}ni=0 c ann (a). There exists a′ є R such that:
a = a a΄,
and
a΄ a i = 0
i = 0, 1…n
4.Proofs
In this section we prove our main results.
Proof of theorem (3.1)
Assume that <a> is flat in R, in order to prove that <a> is flat in R [ x],
we assume that ag =0, where
n
g = Σ aixi є R [x] ,
(3)
i=0
this implies that a ai = 0, for all i = 0,1…n see [2,p179]
since <a> is flat in R, there exist a i΄ є R see [1] such that:
and
a =aai΄,
ai΄ai= 0 .
i = 0,1…n
i = 0,1…n
(4)
(5 )
ICITNS 2003 International Conference on Information Technology and Natural Sciences
n
Choose a΄ =π ai ́, it can be checked easily that:
i=0
a = aa΄,
(6)
aa΄= 0 .
(7)
and
Conversely: Assume that <a> is flat in R [x], we want to prove that <a> is flat in R.
n
Let ab = 0, since <a> is flat in R [x] , there exists g =Σ aixi є R [x] such that:
i=0
a = ag ,
(8)
gb = 0.
(9)
and
It follows from [2,p179] that a =a aο and aο b = 0,therefore
<a> is flat in R.
Proof of theorem (3.2)
Assume that <a> is flat in R. To prove that <a> is flat in R [x, y],
n n
we assume that
ag = 0 where
g =Σ Σ aij xi yj є R [x, y]
j=0 i=0
it follows directly that
a aij =0
for all i,j =0,1…n
(10 )
Since <a> is flat in R, there exist a΄ij є R such that:
and
a=aa΄ij ,
a΄ijaij=0 .
n
Choose a΄= π aij
i,j=0
we get a=aa,΄
i,j=0,1,…. n
(11)
i,j=0,1,…. n
(12)
(13)
(14)
and
a΄g=0
(15)
By straightforward manipulations, we can show that if<a> is flat is R [x, y] then it is
flat in R.
ICITNS 2003 International Conference on Information Technology and Natural Sciences
Proof of theorem (3.3)
The proof of this theorem follows directly from the definition of
R [x1, x2…xn]=R [x1…x n-1] [xn] see [3,p.293] and by using induction on theorem
(3.2).
Proof of corollary (3.1)
The proof of this theorem follows directly from theorems (3.1), (3.2) and (3.3).
Proof of theorem (3.4)
n
To show that (i) implies (ii), assume that {ai}i=0 c ann (a), it follows from [4] that
aia =0,
i=0,1… n
since <a> is flat in R [x], it follows from theorem (3.1 ) that <a> is flat in R.
Hence, there exist a΄iєR such that
a=aa΄i,
and
a΄ia=0 . i =0,1…n
Next, choose a΄=a0΄a1΄…an΄, it is clear that a=aa΄and a΄ai=0
i=0,1…n.
n
Now t, to show that (ii) implies (i), assume that ag =0 where g= Σ aixi єR [x],
i=0
it follows that aai=0 i=0,1…n.
n
This means that {ai}i=0 c ann (a)
Using (ii) it follows that there exist a ΄єR such that
a=aa΄ and a΄ai=0 i=0,1…n,
And so a΄g=0 which implies that <a> is flat in R [x].
REFERANCES
(1) Taha, s.: flat ideals in commutative rings. (A ph.D.Thesis).
(2) I.N.Herestein, Abstract algebra, Macmillan publishing company, New
York, 1990.
(3) A.P.Hillman and G.W Alexanderson, Abstract algebra, PWS publishing
Company, Boston, MA, 1993
(4) Taha, on projective ideals in commutative rings, to appear in institute of
Mathematics and computer science. Kolkata-India
ICITNS 2003 International Conference on Information Technology and Natural Sciences
ICITNS 2003 International Conference on Information Technology and Natural Sciences