Dictionnaire des mathematiques by Collectif

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9). Let I be a nonzero ideal in a unique factorization domain A. Then prin,(II-l) = A. PROOF. We can take x E A such that xA = prin,I. 7) we have that II-1= Ix-'. Let z be any nonzero element in A such that II-' C z A ; then Ix-l C z A and hence I C xzA; consequently xA = prin,ICxzA and hence zA = A. Thus A is the only nonzero principal ideal in A containing II-', and hence prinA(II-') = A. $1. lo). Let I be a nonzero ideal in a unique factorization domain A , and let 0 # y E A. Then: Iy-' C II-' 0 I C y A o prin,I C yA.

Again since x i # / x j EB Q , S dominates BQ , and x j , / x j is a unit in S , we get that x j , / x j is ~ consequently x i / x j , = a unit in BQ and hence x j l x j r B,; ( x i / x j ) ( x j / x j ,E) B , for all i E I and hence B' C B,; since S dominates B , and Q' = B' n M ( S ) , we get that Q' = B' n M(B,) and hence Bbs C B, . 3). 1). Then 'UI(A;(x&,) is a semimodel of K'IA. If I is a finite set then m ( A ;(xi)ip,) is a complete model of K'IA. PROOF. 2). T o prove the second assertion assume that I is a finite set.

4). 6)). Let J be a nonzero principal ideal in R such that ( R , J ) is unresolved. Then (E2(R,J ) is a finite set. PROOF. T h e assertion is obvious if dim R # 3. So assume that = 3. Since ( R , J ) is unresolved, we have that J = P Y1 ... P E n dim R $ 1 . , P, are distinct nonzero principal prime ideals in R. If n > 1 then {SE %(R): dim S = 2 and P, P, C R n M ( S ) } is a finite set and it contains E2(R,J ) , and hence E2(R,J ) is a finite set. So also assume that n = 1, and let P = P I . Then E2(R,J ) = E2(R,P).

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