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单词 CompleteRingOfQuotientsOfReducedCommutativeRings
释义

complete ring of quotients of reduced commutative rings


There is a characterization of complete ring of quotients of reduced commutative rings. Let A be a reduced (http://planetmath.org/ReducedRing) commutative ring, then if B is an overring of A and if for any element bB\\{0} there is an aA such that abA\\{0}, then B is said to be a rational extension of A. See how similar this is with the definition of essential extensionPlanetmathPlanetmath in the category of rings, obviously all rational extensions of reduced commutative rings are also essential extensions. Furthermore there is a maximum (upto A-isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath) rational extension of A and this is in fact the complete ring of quotients of A.

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更新时间:2025/5/4 14:14:07