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Tag Archives: field of fractions
Commutative Algebra 39
Integrality Throughout this article, A is a subring of B; we will also call B a ring extension of A. Definition. An element is said to be integral over A if we can find (where ) such that in B. For example, is integral over since … Continue reading
Posted in Advanced Algebra
Tagged field of fractions, finite extensions, integral closure, integral extensions, normal domains, rings, UFDs
4 Comments
Commutative Algebra 22
Localization Recall that given an integral domain, there is a canonical way to construct the “smallest field containing it”, its field of fractions. Here, we will generalize this construction to arbitrary rings. We let A be a fixed ring throughout. Definition. … Continue reading
Posted in Advanced Algebra
Tagged field of fractions, ideals, localization, universal properties
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Commutative Algebra 17
Field of Fractions Throughout this article, A denotes an integral domain (which may not be a UFD). Definition. The field of fractions of A is an embedding of A into a field K, such that every element of K can be … Continue reading
Posted in Advanced Algebra
Tagged field of fractions, gauss lemma, krull dimension, primes, principal ideal domains, UFDs, unique factorisation
6 Comments