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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