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Is the ring of all algebraic integers coherent? Here is the definition of a coherent ring.

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Yes. In fact every finitely generated ideal is principal.

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en.wikipedia.org/wiki/B%C3%A9zout_domain –  user29743 Nov 20 '12 at 22:16
    
Dear Countinghaus, How do you prove that the ring of all algebraic integers is a Bezout domain? –  Makoto Kato Nov 20 '12 at 22:18
    
This is a little gross and there's probably an easier argument, but given a finitely generated ideal there is an ideal of an honest number field generated by the same generators; this ideal becomes principal after a finite extension (e.g. by capitulation of ideals in the Hilbert class field) and the generator of that new ideal is also a generator for your original ideal. –  user29743 Nov 20 '12 at 22:24
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