Commutative algebra Localization Concept
Multiplicative set
Summary
A multiplicative set in a commutative ring is a subset that contains and is closed under multiplication. These sets are exactly the sets of elements that one may turn into units by localization. The complement of a prime ideal and the set of powers of one element are the two main examples.
Where this leads
10 pages build on this one.
- Regular local rings are UFDs
- Total ring of fractions
- Localization of a Noetherian ring is Noetherian
- Nagata's criterion for factoriality
- Localization
- Localization at a prime ideal
- Localization of a space
- Localization of a module
- Prime ideals of a localization
- Nagata's example of a Noetherian ring of infinite dimension
Further reading
About this page
| Tag | AN9SKEZ3VY |
|---|---|
| URI | m2://AN9SKEZ3VY |
| Kind | concept |
| Writing | W0 stub |
| MSC | 13B30, 13A99 |
| Same as | |
| Changed | 2 November 2026 |
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Used by
- Total ring of fractions
- Localization of a Noetherian ring is Noetherian
- Nagata's criterion for factoriality
- Localization
- Localization at a prime ideal
- Localization of a module
- Prime ideals of a localization
- Nagata's example of a Noetherian ring of infinite dimension