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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.

##### MSC:
 13-XX Commutative algebra
##### Keywords:
integrally closed domains of integrity; arithmetic
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