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Beiträge zur Arithmetik kommutativer Integritätsbereiche. I: Multiplikationsringe, ausgezeichnete Idealsysteme und Kroneckersche Funktionalringe. (German) Zbl 0015.00203
The present paper applies the theory of general non-archimedean valuations and the corresponding evaluation rings (e. r.) previously discussed by the author [J. Reine Angew. Math. 167, 160–196 (1932; Zbl 0004.09802)] to the arithmetic of arbitrary integrally closed domains of integrity \(\mathfrak I\). The result that \(\mathfrak I\) is the intersection \(\Delta\mathfrak B_\tau\), of all e. r.’s \(\mathfrak B_\tau\) where \(\mathfrak K\supset\mathfrak B_\tau\supset\mathfrak I\) and \(\mathfrak K\) is the quotient field of \(\mathfrak I\) and the simple arithmetic of \(\mathfrak B_\tau\) are used to prove theorems about \(\mathfrak I\). For example, Kronecker’s theorem for polynomials with coefficients in \(\mathfrak I\) is proved in this way.
The main concern of the paper is the investigation of extension ideals \(\mathfrak a'\) of \(\mathfrak a\) (any fractional ideal), or of \('\) operations such that (1) \(\mathfrak I' = \mathfrak I\), (2) \(\mathfrak a'\supseteq \mathfrak a\); \(\mathfrak a\supseteq \mathfrak b\) implies \(\mathfrak a'\supseteq \mathfrak b'\), (3) \((\mathfrak a')' = \mathfrak a'\), (4) \((\mathfrak a + \mathfrak b)' = (\mathfrak a' + \mathfrak b')'\), (5) \((\mathfrak a\mathfrak b)' = (\mathfrak a' \mathfrak b')\), (6) \((\mathfrak a' \cap \mathfrak b')' = \mathfrak a' \cap \mathfrak b'\), (7) \((a)' = (a)\); \((a) \mathfrak a' = ((a)\mathfrak a)'\).
If \((\mathfrak a' \mathfrak b')' \subseteq (\mathfrak a' \mathfrak c')'\) for a finite \(\mathfrak a\) implies that \(\mathfrak b' \subseteq \mathfrak c'\), the \('\) operation is called arithmetically useful (a. u.). Such an operation defines a functional ring \(\tilde{\mathfrak M'}\) as the set of rational functions \(\alpha = \frac{(a_0x^m + \ldots + a_m)}{(b_0x^n + \ldots + b_n)}\), \(a_i, b_i\in \mathfrak K\) such that \((a_0, \ldots, a_m)' \subseteq (b_0, \ldots, b_n)'\). \(\tilde{\mathfrak M'}\cap \mathfrak K = \mathfrak I\) and every finite ideal in \(\tilde{\mathfrak M'}\) is principal.
Any a. u. \('\) operation is shown to be equivalent, i. e. has the same functional ring as a \(w\)-operation defined by \(\mathfrak a_w = \Delta \mathfrak a\mathfrak B_\sigma\), where \(\{\mathfrak B_\sigma\}\) is a set of e. r.’s for which \(\Delta\mathfrak B_\sigma = \mathfrak I\). If this is the complete set of e. r.’s containing \(\mathfrak I\) the corresponding operation, \(b\) is equivalent to the integral closure \(a\) of \(\mathfrak a\) as defined by H. Prüfer [J. Reine Angew. Math. 168, 1–36 (1932; Zbl 0004.34001)] and the corresponding functional ring \(\tilde{\mathfrak M}\) is contained in all \(\tilde{\mathfrak M'}\). The latter may be obtained also as quotient rings in the sense of Grell [Math. Ann. 97, 490–523 (1927; JFM 53.0116.03)] of \(\tilde{\mathfrak M}\). If the van der Waerden-Artin closure \(\mathfrak a_v = (\mathfrak a^{-1})^{-1}\) is a. u., its \(\tilde{\mathfrak M}_v\) contains all other \(\tilde{\mathfrak M'}\).
These results also indicate the nature of the problem of determining all e. r.’s \(\mathfrak B_\tau\supset \mathfrak I\). For since \(\tilde{\mathfrak M}\) is a multiplication ring (every finite ideal reversible) its e. r.’s are the rings \(\tilde{\mathfrak M}_{\mathfrak p}\), the sets \(\alpha/\beta\) where \(\alpha,\beta\in\mathfrak M\) and \(\beta\notin\mathfrak p\) prime ideal in \(\tilde{\mathfrak M}\). Furthermore there is a (1-1) correspondence between these e. r.’s and those containing \(\mathfrak I\). Equivalence of \('\) operations does not imply identity. The question whether the particular operations \(a\) and \(b\) are the same is investigated and a partial answer obtained. Other methods of obtaining a. u. operations and also the theory in the special case of rings satisfying the divisor chain condition are discussed.

13-XX Commutative algebra
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