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Abbreviations
- D :
-
an integral domain
- I (D):
-
the set of the fractional ideals ofD formed in the quotient field ofD, closed under the four operations ., +, :, ∩
- S (D):
-
the multiplicative semigroup with division of the arithmetical equivalence classes ofI (D)
- \(\mathfrak{O}\) :
-
a noetherian ring
- T (\(\mathfrak{O}\)):
-
the multiplicative semigroup with division of the weak equivalence classes of the fractional ideals of\(\mathfrak{O}\)
- G (\(\mathfrak{O}\)):
-
the multiplicative group of all invertible\(\mathfrak{O}\)-ideals with order\(\mathfrak{O}\)
- \(\mathfrak{O}'\) :
-
an\(\mathfrak{O}\)-order
- I (\(\left( {{{\mathfrak{O}'} \mathord{\left/ {\vphantom {{\mathfrak{O}'} \mathfrak{O}}} \right. \kern-\nulldelimiterspace} \mathfrak{O}}} \right)\)):
-
the family of all ideals\(\mathfrak{a}\) ofI(\(\left( \mathfrak{O} \right)\)) such that\(\mathfrak{a} \mathfrak{O}' \in G\left( {\mathfrak{O}'} \right)\) ∈G(\(\mathfrak{a} \mathfrak{O}' \in G\left( {\mathfrak{O}'} \right)\))
- T (\(\left( {{{\mathfrak{O}'} \mathord{\left/ {\vphantom {{\mathfrak{O}'} \mathfrak{O}}} \right. \kern-\nulldelimiterspace} \mathfrak{O}}} \right)\)):
-
the family of weak equivalence classes of elements ofI (\(\left( {{{\mathfrak{O}'} \mathord{\left/ {\vphantom {{\mathfrak{O}'} \mathfrak{O}}} \right. \kern-\nulldelimiterspace} \mathfrak{O}}} \right)\))
- J (\(\left( {{{\mathfrak{O}'} \mathord{\left/ {\vphantom {{\mathfrak{O}'} \mathfrak{O}}} \right. \kern-\nulldelimiterspace} \mathfrak{O}}} \right)\)):
-
the family of all\(\mathfrak{O}\)-submodules ≠ 0 of\(\mathfrak{O}'\)
- U (\(\left( {{{\mathfrak{O}'} \mathord{\left/ {\vphantom {{\mathfrak{O}'} \mathfrak{O}}} \right. \kern-\nulldelimiterspace} \mathfrak{O}}} \right)\)):
-
the family of all ideals\(\mathfrak{a}\) ofJ (\(\left( {{{\mathfrak{O}'} \mathord{\left/ {\vphantom {{\mathfrak{O}'} \mathfrak{O}}} \right. \kern-\nulldelimiterspace} \mathfrak{O}}} \right)\)) such that\(\mathfrak{a}\mathfrak{O}' = \mathfrak{O}'\)
- V (\(\left( {{{\mathfrak{O}'} \mathord{\left/ {\vphantom {{\mathfrak{O}'} \mathfrak{O}}} \right. \kern-\nulldelimiterspace} \mathfrak{O}}} \right)\)):
-
the multiplicative semigroup with division of the weak equivalence classes contained inU (\(\left( {{{\mathfrak{O}'} \mathord{\left/ {\vphantom {{\mathfrak{O}'} \mathfrak{O}}} \right. \kern-\nulldelimiterspace} \mathfrak{O}}} \right)\))
- ℒ:
-
the ring of the rational integers
- Q :
-
the rational field
- \(\mathfrak{O}_\mathfrak{p} \) :
-
the local ring of all elementsx/ν (x ∈\(\mathfrak{O}\),ν ∈\(\mathfrak{O}\),ν ∉\(\mathfrak{p}\)) belonging to the prime ideal\(\mathfrak{p}\) of the integral domain\(\mathfrak{O}\)
- \(\mathfrak{a}_\mathfrak{p} \) :
-
the local extension a\(\mathfrak{a} \mathfrak{O}_\mathfrak{p} \) of a member\(\mathfrak{a}\) ofI (\(\mathfrak{O}\)) to a member ofI (\(\mathfrak{O}_\mathfrak{p} \))
- N (\(\mathfrak{a}\)):
-
a non-negative integer with the property that\(\mathfrak{a}^{N\left( \mathfrak{a} \right)} \) is invertible, but\(\mathfrak{a}^{N\left( \mathfrak{a} \right) - 1} \) is not invertible.
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This work was supported in part by the Office of Naval Research at the California Institute of Technology and by the National Science Foundation at the California Institute of Technology (including the 1960 Group Theory Institute) and at the University of Notre Dame (1961 Symposium on Number Theory and Algebraic Geometry).
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Dade, E.C., Taussky, O. & Zassenhaus, H. On the theory of orders, in particular on the semigroup of ideal classes and genera of an order in an algebraic number field. Math. Ann. 148, 31–64 (1962). https://doi.org/10.1007/BF01438389
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DOI: https://doi.org/10.1007/BF01438389