Tag Archives: field of fractions

Commutative Algebra 39

Posted on April 27, 2020 by limsup

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 , , , , , , | 4 Comments

Commutative Algebra 22

Posted on April 9, 2020 by limsup

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 →

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Commutative Algebra 17

Posted on April 4, 2020 by limsup

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 , , , , , , | 6 Comments