What does regular ring mean?
Definitions for regular ring
reg·u·lar ring
This dictionary definitions page includes all the possible meanings, example usage and translations of the word regular ring.
Wikipedia
Regular ring
In commutative algebra, a regular ring is a commutative Noetherian ring, such that the localization at every prime ideal is a regular local ring: that is, every such localization has the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. The origin of the term regular ring lies in the fact that an affine variety is nonsingular (that is every point is regular) if and only if its ring of regular functions is regular. For regular rings, Krull dimension agrees with global homological dimension. Jean-Pierre Serre defined a regular ring as a commutative noetherian ring of finite global homological dimension. His definition is stronger than the definition above, which allows regular rings of infinite Krull dimension. Examples of regular rings include fields (of dimension zero) and Dedekind domains. If A is regular then so is A[X], with dimension one greater than that of A. In particular if k is a field, the polynomial ring k [ X 1 , … , X n ] {\displaystyle k[X_{1},\ldots ,X_{n}]} is regular. This is Hilbert's syzygy theorem. Any localization of a regular ring is regular as well. A regular ring is reduced but need not be an integral domain. For example, the product of two regular integral domains is regular, but not an integral domain.
Numerology
Chaldean Numerology
The numerical value of regular ring in Chaldean Numerology is: 6
Pythagorean Numerology
The numerical value of regular ring in Pythagorean Numerology is: 4
Translation
Find a translation for the regular ring definition in other languages:
Select another language:
- - Select -
- 简体中文 (Chinese - Simplified)
- 繁體中文 (Chinese - Traditional)
- Español (Spanish)
- Esperanto (Esperanto)
- 日本語 (Japanese)
- Português (Portuguese)
- Deutsch (German)
- العربية (Arabic)
- Français (French)
- Русский (Russian)
- ಕನ್ನಡ (Kannada)
- 한국어 (Korean)
- עברית (Hebrew)
- Gaeilge (Irish)
- Українська (Ukrainian)
- اردو (Urdu)
- Magyar (Hungarian)
- मानक हिन्दी (Hindi)
- Indonesia (Indonesian)
- Italiano (Italian)
- தமிழ் (Tamil)
- Türkçe (Turkish)
- తెలుగు (Telugu)
- ภาษาไทย (Thai)
- Tiếng Việt (Vietnamese)
- Čeština (Czech)
- Polski (Polish)
- Bahasa Indonesia (Indonesian)
- Românește (Romanian)
- Nederlands (Dutch)
- Ελληνικά (Greek)
- Latinum (Latin)
- Svenska (Swedish)
- Dansk (Danish)
- Suomi (Finnish)
- فارسی (Persian)
- ייִדיש (Yiddish)
- հայերեն (Armenian)
- Norsk (Norwegian)
- English (English)
Word of the Day
Would you like us to send you a FREE new word definition delivered to your inbox daily?
Citation
Use the citation below to add this definition to your bibliography:
Style:MLAChicagoAPA
"regular ring." Definitions.net. STANDS4 LLC, 2024. Web. 6 May 2024. <https://www.definitions.net/definition/regular+ring>.
Discuss these regular ring definitions with the community:
Report Comment
We're doing our best to make sure our content is useful, accurate and safe.
If by any chance you spot an inappropriate comment while navigating through our website please use this form to let us know, and we'll take care of it shortly.
Attachment
You need to be logged in to favorite.
Log In