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A Characterization Of Krull Rings With Zero Divisors

By Franz Halter-Koch and M T (r


. It is proved that a Marot ring is a Krull ring if and only if its monoid of regular elements is a Krull monoid. It was first noticed by L. Skula [7] that a domain R is a Krull domain if and only 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 zero divisors as treated in [4]. All rings in this note are commutative and possess a unit element. If R is a ring, we denote by R ffl the monoid of regular elements of R , by R \Theta the group of invertible elements of R and by T (R) a total quotient ring of R ; clearly, T (R) ffl = T (R) \Theta . For a prime ideal P of R , we set R (P ) = (R ffl n P ) \Gamma1 R ae T (R) . Throughout, we shall assume that R is a Marot ring, 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 cond..

Year: 1993
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