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dbr:Dimension_theory_(algebra)
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Dimension theory (algebra) نظرية البعد (الجبر) 次元論 (代数学)
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In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme). The need of a theory for such an apparently simple notion results from the existence of many definitions of the dimension that are equivalent only in the most regular cases (see Dimension of an algebraic variety). A large part of dimension theory consists in studying the conditions under which several dimensions are equal, and many important classes of commutative rings may be defined as the rings such that two dimensions are equal; for example, a regular ring is a commutative ring such that the homological dimension is equal to the Krull dimension. 数学において、次元論（じげんろん、英: dimension theory）は可換環論の一分野であり、可換環の次元の概念や、より一般にスキームのそれを研究する分野である。 理論はアフィン環、すなわち体上有限生成多元環である整域に対しては、はるかに単純である。ネーターの正規化定理により、そのような環のクルル次元は基礎体上の超越次数であり、理論は代数幾何学と並行して進む。を参照。一般的な理論は幾何学的でなくなる傾向がある。特に、ネーター的でない環に対して知られていることはほとんどない。（Kaplansky の commutative rings は非ネーターのケースに詳しい。）今日、標準的なアプローチは本質的にブルバキとEGAのアプローチである。これは次数付き加群を本質的に使い、他のものの中で射影多様体の次数の一般化であるの役割を強調する。このアプローチでは、クルルの単項イデアル定理は系として現れる。 この記事を通して、 は環のクルル次元を表し、 は素イデアルのクルル次元（すなわちその素イデアルにおける局所化のクルル次元）を表す。 في الرياضيات, نظرية البعد هي أحد فروع الجبر التبادلي التي تدرس فكرة البعد للحلقة التبادلية, وبتمديدها بواسطة المخطط scheme.
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Lemma
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In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme). The need of a theory for such an apparently simple notion results from the existence of many definitions of the dimension that are equivalent only in the most regular cases (see Dimension of an algebraic variety). A large part of dimension theory consists in studying the conditions under which several dimensions are equal, and many important classes of commutative rings may be defined as the rings such that two dimensions are equal; for example, a regular ring is a commutative ring such that the homological dimension is equal to the Krull dimension. The theory is simpler for commutative rings that are finitely generated algebras over a field, which are also quotient rings of polynomial rings in a finite number of indeterminates over a field. In this case, which is the algebraic counterpart of the case of affine algebraic sets, most of the definitions of the dimension are equivalent. For general commutative rings, the lack of geometric interpretation is an obstacle to the development of the theory; in particular, very little is known for non-noetherian rings. (Kaplansky's Commutative rings gives a good account of the non-noetherian case.) Throughout the article, denotes Krull dimension of a ring and the height of a prime ideal (i.e., the Krull dimension of the localization at that prime ideal.) Rings are assumed to be commutative except in the last section on dimensions of non-commutative rings. 数学において、次元論（じげんろん、英: dimension theory）は可換環論の一分野であり、可換環の次元の概念や、より一般にスキームのそれを研究する分野である。 理論はアフィン環、すなわち体上有限生成多元環である整域に対しては、はるかに単純である。ネーターの正規化定理により、そのような環のクルル次元は基礎体上の超越次数であり、理論は代数幾何学と並行して進む。を参照。一般的な理論は幾何学的でなくなる傾向がある。特に、ネーター的でない環に対して知られていることはほとんどない。（Kaplansky の commutative rings は非ネーターのケースに詳しい。）今日、標準的なアプローチは本質的にブルバキとEGAのアプローチである。これは次数付き加群を本質的に使い、他のものの中で射影多様体の次数の一般化であるの役割を強調する。このアプローチでは、クルルの単項イデアル定理は系として現れる。 この記事を通して、 は環のクルル次元を表し、 は素イデアルのクルル次元（すなわちその素イデアルにおける局所化のクルル次元）を表す。 في الرياضيات, نظرية البعد هي أحد فروع الجبر التبادلي التي تدرس فكرة البعد للحلقة التبادلية, وبتمديدها بواسطة المخطط scheme.
dbp:mathStatement
The maximal ideal contains elements , d = Krull dimension of R, such that, for any i, any prime ideal containing has height .
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