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| {{mergefrom|Height (ring theory)|date=July 2013}}
| | Dr. Paul L. Gerhardt offers customized [http://ezinearticles.com/?Whats-the-Point-of-Leadership-Training-Programs?&id=8679933 leadership skills training] for managers and supervisors that wish to accomplish more at a professional level, increase their productivity and gain the trust and loyalty of their employees. To help people enhance their leadership capabilities, Dr. Paul L. Gerhardt uses various methods such as keynote speaking and training workshops. As a trusted published author and leadership adviser, “The Organizational Doctor” has managed to aid companies and various organizations to identify and train their talents into competent and competitive employees. |
| In [[commutative algebra]], the '''Krull dimension''' of a [[commutative ring]] ''R'', named after [[Wolfgang Krull]], is the supremum of the lengths of all chains of [[prime ideals]]. The Krull dimension need not be finite even for a [[Noetherian ring]]. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the [[deviation of a poset|deviation]] of the poset of submodules.
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| The Krull dimension has been introduced to provide an algebraic definition of the [[dimension of an algebraic variety]]: the dimension of the [[affine variety]] defined by an ideal ''I'' in a [[polynomial ring]] ''R'' is the Krull dimension of ''R''/''I''. | | It is important to know how to nurture the talent in a company and ensure the work environment is inspirational. The leadership training programs Dr. Paul L. Gerhardt organizes not only allows managers to see the importance of a healthy working environment for the productivity and overall success of a company, but also help them develop essential leadership skills. Dr. Gerhardt is renowned for its services, workshops, articles and books and those interested in appealing to his services can visit paulgerhardt. |
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| A field ''k'' has Krull dimension 0; more generally, ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>] has Krull dimension ''n''. A [[principal ideal domain]] that is not a field has Krull dimension 1. A [[local ring]] has Krull dimension 0 if and only if every element of its maximal ideal is nilpotent.
| | com and read more about what he can offer and contact him. Services Consulting – dysfunction at the working place is not only a possibility, it is a reality. Dr. Gerhardt specializes in offering consulting services in order to identify the causes of an unhealthy working environment and takes an innovative approach to curing those issues. With focus on working relationships and individual attitudes, he tries to motivate talents and align their aspirations with the purpose of the company they work with. |
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| == Explanation ==
| | Leadership coaching - Dr. Gerhardt has a Master’s Degree in Organizational Leadership and a Ph.D. in Business Management and Organizational Behaviors. Therefore, he is considered an expert in coaching inexperienced leaders that can rely on his knowledge and experience to become the next powerful business leaders of the future. His leadership skills training focuses on an open teaching style, individual coaching and skill-building exercises. Seminars - Dr. |
| We say that a chain of prime ideals of the form
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| <math>\mathfrak{p}_0\subsetneq \mathfrak{p}_1\subsetneq \ldots \subsetneq \mathfrak{p}_n</math>
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| has '''length n'''. That is, the length is the number of strict inclusions, not the number of primes; these differ by 1. We define the '''Krull dimension''' of <math>R</math> to be the supremum of the lengths of all chains of prime ideals in <math>R</math>.
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| Given a prime <math>\mathfrak{p}</math> in ''R'', we define the '''[[height (ring theory)|height]]''' of <math>\mathfrak{p}</math>, written <math>\operatorname{ht}(\mathfrak{p})</math>, to be the supremum of the lengths of all chains of prime ideals contained in <math>\mathfrak{p}</math>, meaning that <math>\mathfrak{p}_0\subsetneq \mathfrak{p}_1\subsetneq \ldots \subsetneq \mathfrak{p}_n\subseteq \mathfrak{p}</math>. In other words, the height of <math>\mathfrak{p}</math> is the Krull dimension of the [[localization of a ring|localization]] of ''R'' at <math>\mathfrak{p}</math>. A prime ideal has height zero if and only if it is a [[minimal prime ideal]]. The Krull dimension of a ring is the supremum of the heights of all maximal ideals, or those of all prime ideals.
| | Gerhardt also holds the position of professional educator and organizes quality seminars at the workplace. The aim of the seminars is to engage employees and help them develop certain skills such as public speaking skills and other essential sets of skills that will increase their productivity and potential. [http://berojgarjago.com/article/article.php?id=43052 leadership workshops] – leadership involves more than supervision and maintaining the status quo in the company. |
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| In a Noetherian ring, every prime ideal has finite height. Nonetheless,
| | Dr. Gerhardt organizes ingenious leadership workshops to help managers and supervisors adopt and develop a mindset worthy of a true leader. This involves helping them find ways to obtain the best out of their employees, earn their trust and keep them motivated. Participants to the workshops are generally taught supervision basics and how to acquire certain skills and tools necessary for a good supervision such as using non-intrusive ways of control and giving encouragements to the staff when needed. |
| Nagata gave an example of a Noetherian ring of infinite Krull dimension.<ref>Eisenbud, D. ''Commutative Algebra'' (1995). Springer, Berlin. Exercise 9.6.</ref> A ring is called '''[[catenary ring|catenary]]''' if any inclusion <math>\mathfrak{p}\subset \mathfrak{q}</math> of prime ideals can be extended to a maximal chain of prime ideals between <math>\mathfrak{p}</math> and <math>\mathfrak{q}</math>, and any two maximal chains between <math>\mathfrak{p}</math>
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| and <math>\mathfrak{q}</math> have the same length. Nagata gave an example of a Noetherian ring which is not catenary.<ref>Matsumura, H. ''Commutative Algebra'' (1970). Benjamin, New York. Example 14.E.</ref>
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| ==Krull dimension and schemes==
| | Dr. Paul L. Gerhardt is renowned on the dedicated market for its innovative approach to leadership training. Those interested in increasing the productivity of their company and explore the potential of their employees or wishing to solve work related issues without losing their talents can resort to the services mentioned above. |
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| It follows readily from the definition of the [[spectrum of a ring]] Spec(''R''), the space of prime ideals of ''R'' equipped with the Zariski topology, that the Krull dimension of ''R'' is equal to the dimension of its spectrum as a topological space, meaning the supremum of the lengths of all chains of irreducible closed subsets. This follows immediately from the [[Galois connection]] between ideals of ''R'' and closed subsets of Spec(''R'') and the observation that, by the definition of Spec(''R''), each prime ideal <math>\mathfrak{p}</math> of ''R'' corresponds to a generic point of the closed subset associated to <math>\mathfrak{p}</math> by the Galois connection.
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| ==Examples==
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| * The dimension of a [[polynomial ring]] over a field ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>] is the number of variables ''n''. In the language of [[algebraic geometry]], this says that the affine space of dimension ''n'' over a field has dimension ''n'', as expected. In general, if ''R'' is a [[Noetherian ring|Noetherian]] ring of dimension ''n'', then the dimension of ''R''[''x''] is ''n'' + 1. If the Noetherian hypothesis is dropped, then ''R''[''x''] can have dimension anywhere between ''n'' + 1 and 2''n'' + 1.
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| * The ring of integers '''Z''' has dimension 1. More generally, any [[principal ideal domain]] that is not a field has dimension 1.
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| * An [[integral domain]] is a field if and only if its Krull dimension is zero. [[Dedekind domain]]s that are not fields (for example, [[discrete valuation ring]]s) have dimension one. A [[Noetherian ring|Noetherian]] ring is [[Artinian ring|Artinian]] if and only if its Krull dimension is 0.
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| * An [[integral extension]] of a ring has the same dimension as the ring does.
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| * Let ''R'' be an algebra over a field ''k'' that is an integral domain. Then the Krull dimension of ''R'' is less than or equal to the transcendence degree of the field of fractions of ''R'' over ''k''.<ref>http://mathoverflow.net/questions/79959/krull-dimension-transcendence-degree</ref> The equality holds if ''R'' is finitely generated as algebra (for instance by the [[noether normalization lemma]]).
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| * Let ''R'' be a noetherian ring, ''I'' an ideal and <math>\operatorname{gr}_I(R) = \oplus_0^\infty I^k/I^{k+1}</math> be the [[associated graded ring]] (geometers call it the ring of the [[normal cone]] of ''I''.) Then <math>\operatorname{dim} \operatorname{gr}_I(R)</math> is the supremum of the heights of maximal ideals of ''R'' containing ''I''.<ref>{{harvnb|Eisenbud|2004|loc=Exercise 13.8}}</ref>
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| * A Noetherian local ring is called a [[Cohen–Macaulay ring]] if its dimension is equal to its [[Depth (ring theory)|depth]]. A [[regular local ring]] is an example of such a ring.
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| ==Krull dimension of a module==
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| If ''R'' is a commutative ring, and ''M'' is an ''R''-module, we define the Krull dimension of ''M'' to be the Krull dimension of the quotient of ''R'' making ''M'' a [[faithful module]]. That is, we define it by the formula:
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| :<math>\operatorname{dim}_R M := \operatorname{dim}( R/\operatorname{Ann}_R(M))</math>
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| where Ann<sub>''R''</sub>(''M''), the [[annihilator (ring theory)|annihilator]], is the kernel of the natural map R → End<sub>''R''</sub>(M) of ''R'' into the ring of ''R''-linear endomorphisms of ''M''.
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| In the language of [[Scheme (mathematics)|schemes]], finitely generated modules are interpreted as [[coherent sheaves]], or generalized finite rank [[vector bundles]].
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| ==Krull dimension for non-commutative rings==
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| The Krull dimension of a module over a possibly non-commutative ring is defined as the [[deviation of a poset|deviation]] of the poset of submodules ordered by inclusion. For commutative Noetherian rings, this is the same as the definition using chains of prime ideals.<ref>McConnell, J.C. and Robson, J.C. ''Noncommutative Noetherian Rings'' (2001). Amer. Math. Soc., Providence. Corollary 6.4.8.</ref> The two definitions can be different for commutative rings which are not Noetherian.
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| == See also ==
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| *[[Dimension theory (algebra)]]
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| *[[Regular local ring]]
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| *[[Hilbert function]]
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| *[[Krull's principal ideal theorem]]
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| ==Notes==
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| {{reflist}}
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| ==Bibliography==
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| * [[Irving Kaplansky]], ''Commutative rings (revised ed.)'', [[University of Chicago Press]], 1974, ISBN 0-226-42454-5. Page 32.
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| * {{cite book | author1=L.A. Bokhut' | author2=I.V. L'vov | author3=V.K. Kharchenko | chapter=I. Noncommuative rings | editor1-first=A.I. | editor1-last=Kostrikin | editor1-link=A.I. Kostrikin | editor2-first=I.R. | editor2-last=Shafarevich | editor2-link=Igor Shafarevich | title=Algebra II | series=Encyclopaedia of Mathematical Sciences | volume=18 | publisher=[[Springer-Verlag]] | year=1991 | isbn=3-540-18177-6 }} Sect.4.7.
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| * {{Citation | last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=Commutative algebra with a view toward algebraic geometry | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94268-1 | mr=1322960 | year=1995 | volume=150}}
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| [[Category:Commutative algebra]]
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| [[Category:Dimension]]
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Dr. Paul L. Gerhardt offers customized leadership skills training for managers and supervisors that wish to accomplish more at a professional level, increase their productivity and gain the trust and loyalty of their employees. To help people enhance their leadership capabilities, Dr. Paul L. Gerhardt uses various methods such as keynote speaking and training workshops. As a trusted published author and leadership adviser, “The Organizational Doctor” has managed to aid companies and various organizations to identify and train their talents into competent and competitive employees.
It is important to know how to nurture the talent in a company and ensure the work environment is inspirational. The leadership training programs Dr. Paul L. Gerhardt organizes not only allows managers to see the importance of a healthy working environment for the productivity and overall success of a company, but also help them develop essential leadership skills. Dr. Gerhardt is renowned for its services, workshops, articles and books and those interested in appealing to his services can visit paulgerhardt.
com and read more about what he can offer and contact him. Services Consulting – dysfunction at the working place is not only a possibility, it is a reality. Dr. Gerhardt specializes in offering consulting services in order to identify the causes of an unhealthy working environment and takes an innovative approach to curing those issues. With focus on working relationships and individual attitudes, he tries to motivate talents and align their aspirations with the purpose of the company they work with.
Leadership coaching - Dr. Gerhardt has a Master’s Degree in Organizational Leadership and a Ph.D. in Business Management and Organizational Behaviors. Therefore, he is considered an expert in coaching inexperienced leaders that can rely on his knowledge and experience to become the next powerful business leaders of the future. His leadership skills training focuses on an open teaching style, individual coaching and skill-building exercises. Seminars - Dr.
Gerhardt also holds the position of professional educator and organizes quality seminars at the workplace. The aim of the seminars is to engage employees and help them develop certain skills such as public speaking skills and other essential sets of skills that will increase their productivity and potential. leadership workshops – leadership involves more than supervision and maintaining the status quo in the company.
Dr. Gerhardt organizes ingenious leadership workshops to help managers and supervisors adopt and develop a mindset worthy of a true leader. This involves helping them find ways to obtain the best out of their employees, earn their trust and keep them motivated. Participants to the workshops are generally taught supervision basics and how to acquire certain skills and tools necessary for a good supervision such as using non-intrusive ways of control and giving encouragements to the staff when needed.
Dr. Paul L. Gerhardt is renowned on the dedicated market for its innovative approach to leadership training. Those interested in increasing the productivity of their company and explore the potential of their employees or wishing to solve work related issues without losing their talents can resort to the services mentioned above.