summary:It is proved that a Marot ring is a Krull ring if and only if its monoid of regular elements is a Krull monoid

Halter-Koch, Franz

English

Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumFranz Halter-KochA characterization of Krull rings with zero divisorsArchivum Mathematicum, Vol. 29 (1993), No. 1-2, 119--122Persistent URL: http://dml.cz/dmlcz/107472Terms of use:© Masaryk University, 1993Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCHIVUM MATHEMATICUM (BRNO)Tomus 29 (1993), 119 { 122A CHARACTERIZATION OF KRULLRINGS WITH ZERO DIVISORSFranz Halter-KochAbstract. It is proved that a Marot ring is a Krull ring if and only if its monoidof regular elements is a Krull monoid.It was rst noticed by L. Skula [7] that a domain R is a Krull domain if andonly if the multiplicative monoid R n f0g is a Krull monoid (or, equivalently,admits a divisor theory). For independent proofs and historical remarks see [1]and [3].In this note we extend the above-mentioned result to Krull rings with zerodivisors as treated in [4]. All rings in this note are commutative and possess aunit element. If R is a ring, we denote by R the monoid of regular elementsof R , by R the group of invertible elements of R and by T (R) a totalquotient ring of R ; clearly, T (R) = T (R) . For a prime ideal P of R , we setR(P ) = (R n P ) 1R  T (R) . Throughout, we shall assume that R is a Marotring, and we shall use the Marot property in the following form.Lemma. A ring R is a Marot ring if and only if the following condition issatised: (M)  For any two R-submodules M1;M2 of T (R);M1 \ T (R) =M2 \ T (R) 6= ; implies M1 =M2:Proof. By [4], Theorem 7.1, R is a Marot ring if and only if every regular R-submodule of T (R) is generated by its regular elements. Therefore every Marotring satises (M) .Now let R be a ring satisfying (M) and let M  T (R) be a regular R-submodule. Let M0  M be the R-submodule generated by M \ T (R) ; itsatises M0 \ T (R) =M \ T (R) 6= ; , and therefore M0 = M . For the valuation theory of monoids and the theory of Krull monoids we referto [3]. The main result of this note is the following Theorem.1991 Mathematics Subject Classication : 13F05.Key words and phrases: Krull ring, Marot ring, divisor theory, essential valuation, discreterank one valuation ring.Received February 8, 1993. 119120 FRANZ HALTER{KOCHTheorem. Let R be a Marot ring. Then R is a Krull ring if and only if Ris a Krull monoid.Proof of the Theorem (Part 1). Let R be a Krull ring. Then there existsa set  of rank one valuations v : T (R) ! Z[ f1g such that R = fx 2T (R) j v(x)  0 for all v 2 g and, for any x 2 T (R); v(x) = 0 for allbut nitely many v 2  . If v 2  , then v(x) 2 Z for all x 2 T (R) , andv = vjT (R) : T (R) ! Z is a valuation of R . The set fv j v 2 g is adening set of valuations of R , and therefore R is a Krull monoid. The proof of the non-trivial part of the Theorem rests on the following Propo-sition.Proposition. Let R be a Marot ring, v : T (R) !Z an essential valuation ofR , and let P / R be the ideal generated by fx 2 R j v(x) > 0g .i) If n 2 N; x1; : : : ; xn 2 R; 1; : : : ; n 2 R and x = 1x1+  +nxn 2 R ,then v(x)  minfv(x1); : : : ; v(xn)g .ii) P is a prime ideal of R , thenR(P ) \ T (R) = x 2 T (R) j v(x)  0	 ;PR(P ) \ T (R) = x 2 T (R) j v(x) > 0	and R(P ) = x 2 T (R) j v(x) = 0	 :iii) R(P ) is a discrete rank one valuation ring.Proof. i)We may suppose that n  2 and v(x1) = minfv(x1); : : : ; v(xn)g . For2    n , we have x 11 x 2 T (R); v(x 11 x)  0 , and since v is essential forR , there exists an element z 2 R such that v(z) = 0 and zx 11 x 2 R .Putting z = z2  : : :zn 2 R , we obtain v(z) = 0; zx 11 x 2 R for 2    n ,and hence zx 11 x = 1z + nX=2 zx 11 x 2 R ;consequently, 0  v(zx 11 x) =  v(x1) + v(x) , and the assertion follows.ii) By i), we obtain P \R = fx 2 R j v(x) > 0g :For any x; y 2 R; xy 2 P implies 0 < v(xy) = v(x) + v(y) , and sincev(x)  0; v(y)  0 , we conclude v(x) > 0 or v(y) > 0 , i.e. x 2 P or y 2 P .Hence P is a prime ideal by [4], Theorem 7.10.By construction, every x 2 R(P )\T (R) satises v(x)  0; x 2 PR(P )\T (R)implies v(x) > 0 , and x 2 R(P ) implies v(x) = 0 . For the converse, letx 2 T (R) be an element satisfying v(x)  0 . Since v is essential for R , thereA CHARACTERIZATION OF KRULL RINGS WITH ZERO DIVISORS 121exists some z 2 R such that xz 2 R and v(z) = 0 . This implies z =2 P ,and consequently x = xzz 2 R(P ) . If v(x) > 0 , then v(xz) = v(x) > 0 , whencexz 2 P and x 2 PR(P ) . If v(x) = 0 , then x and x 1 both lie in R(P ) ,whence x 2 R(P ) .iii)By [6], Proposition 22, we must prove that PR(P ) is the only regular primeideal of R(P ) , and that it is an invertible ideal.Let t 2 T (R) be an element satisfying v(t) = 1 . By ii), t 2 PR(P ) , and weclaim that PR(P ) = R(P )t . Clearly, it is sucient to prove that PR(P )\T (R) R(P )t . If x 2 PR(P )\T (R) , then v(x) > 0 and hence v(xt 1) = v(x) 1  0 ,which implies xt 1 2 R(P ) and x 2 R(P )t . Being a regular principal ideal,PR(P ) is invertible.If Q/R(P ) is a regular prime ideal and x 2 Q\T (R) , then v(x) > 0 by ii).This implies v(tv(x)x 1) = 0 , hence tv(x)x 1 = e 2 R(P ) and tv(x) = xe 2 Q ,whence t 2 Q and PR(P )  Q . Since (R(P ) nPR(P ))\T (R) = R(P ) , the idealPR(P ) is a maximal regular ideal, and therefore PR(P ) = Q . Proof of the Theorem (Part 2). Let R be a Krull monoid and  the setof essential valuations of R . For v 2  , let Pv / R be the ideal generated byfx 2 R j v(x) > 0g . By the Proposition, Pv is a prime ideal and R(Pv) is adiscrete rank one valuation ring. Therefore it is sucient to prove thatR = \v2R(Pv) ;and every x 2 T (R) lies in R(Pv) for all but nitely many v 2  .By [3], Satz 1,  is a dening set of valuations for R , which means thatR = fx 2 T (R) j v(x)  0 for all v 2 g and, for all x 2 T (R); v(x) = 0for all but nitely many v 2  . By the Proposition, this impliesR = \v2R(Pv) \ T (R);and hence R = \v2R(Pv)by the Lemma; furthermore, if x 2 T (R) , then x 2 R(Pv) for all but nitelymany v 2  . Remark. That the monoid of regular elements of a Krull ring is a Krull monoid,was already observed in [2]. Yet another characterization of Krull rings with zerodivisors was given in [5].122 FRANZ HALTER{KOCHReferences[1] Borewicz, Z. I., Shafarevich, I. R., Number Theory, 3rd ed. (Russian), Moskva 1985.[2] Geroldinger, A., On the arithmetic of certain not integrally closed noetherian domains,Comm. Alg. 19 (1991), 685-698.[3] Halter-Koch, F., Ein Approximationssatz fur Halbgruppen mit Divisorentheorie, Result.Math. 19 (1991), 74-82.[4] Huckaba, J. A., Commutative rings with zero divisors, Marcel Dekker, 1988.[5] Kang, B. G., A characterization of Krull rings with zero divisors, J. Pure Appl. Algebra 72(1991), 33-38.[6] Portelli, D., Spangher,W., Krull rings with zero divisors, Comm. Alg. 11 (1983), 1817-1851.[7] Skula, L., On c-semigroups, Acta Arith. 31 (1976), 247-256.Franz Halter-KochInstitut fur MathematikKarl-Franzens-UniversitatHeinrichstrae 36/IVA-8010 Graz, OSTERREICH

A characterization of Krull rings with zero divisors

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A characterization of Krull rings with zero divisors

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Institute of Mathematics AS CR, v. v. i.