In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules.
In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal.
The concept of local rings was introduced by Wolfgang Krull in 1938 under the name Stellenringe.[1] The English term local ring is due to Zariski.[2]
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Krull, Wolfgang (1938). "Dimensionstheorie in Stellenringen". J. Reine Angew. Math. (in German). 1938 (179): 204. doi:10.1515/crll.1938.179.204. S2CID 115691729.
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Zariski, Oscar (May 1943). "Foundations of a General Theory of Birational Correspondences" (PDF). Trans. Amer. Math. Soc. 53 (3). American Mathematical Society: 490–542 [497]. doi:10.2307/1990215. JSTOR 1990215.
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