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8 - Modules
John Stillwell, University of San Francisco
Book:

Algebraic Number Theory for Beginners

Published online:

28 July 2022

Print publication:

11 August 2022, pp 171-188
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Summary

Modules are like vector spaces, except that their "scalars" are merely from a ring rather than a field. Because of this, modules do not generally have bases. However, we escape the difficulties in the rings of algebraic integers in algebraic number fields, and we can find bases for them with the help of the discriminant. This leads to another property of the latter rings - being integrally closed. In the next chapter we will see that the property of being integrally closed, together with the Noetherian property, is needed to characterize the rings in which unique prime ideal factorization holds.