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{{about|the branch of algebra that studies commutative rings|algebras that are commutative|Commutative algebra (structure)}}
{{Infobox scientist
|name = Walter H. Schottky
|image = Walter Hermann Schottky (1886-1976).jpg
|image_size = 200px
|caption =
|birth_date = {{birth-date|23 July 1886|23 July 1886}}
|birth_place = [[Zürich]], [[Switzerland]]
|death_date = {{death-date|4 March 1976|4 March 1976}}
|death_place = [[Pretzfeld]], [[West Germany]]
|residence = Germany
|citizenship =
|nationality = German
|ethnicity =
|fields = [[Physicist]]
|workplaces = [[University of Jena]]<br />[[University of Würzburg]]<br />[[University of Rostock]]<br />[[Siemens AG|Siemens Research Laboratories]]
|alma_mater = [[University of Berlin]]
|doctoral_advisor = [[Max Planck]]<br />[[Heinrich Rubens]]
|academic_advisors =
|doctoral_students =
|notable_students = [[Werner Hartmann (physicist)|Werner Hartmann]]
|known_for = [[Schottky effect]]<br />[[Schottky barrier]]<br />[[Schottky contact]]<br />[[Schottky anomaly]]<br />[[Screen-grid|Screen-grid vacuum tube]]<br />[[Tetrode]]<br />[[Ribbon microphone]]<br />[[Loudspeaker#Ribbon_and_planar_magnetic_loudspeakers|Ribbon loudspeaker]]<br />[[Field electron emission|Theory of Field emission]]<br />[[Shot noise]]
|author_abbrev_bot =
|author_abbrev_zoo =
|influences =
|influenced =
|awards = [[Hughes medal]] (1936)<br />[[Werner von Siemens Ring]] (1964)
|religion =
|signature = <!--(filename only)-->
|footnotes = His father was the [[mathematician]] [[Friedrich Hermann Schottky]].
}}
'''Walter Hermann Schottky''' (23 July 1886, [[Zürich]], Switzerland – 4 March 1976, [[Pretzfeld]], West Germany) was a German physicist who played a major early role in developing the theory of electron and ion emission phenomena, invented the [[Screen-grid#Screen grid|screen-grid]] [[vacuum tube]] in 1915 and the [[pentode]]{{Citation needed|date=March 2011}} in 1919 while working at [[Siemens AG|Siemens]], co-invented the [[Ribbon microphone]] and [[Loudspeaker#Ribbon_and_planar_magnetic_loudspeakers|Ribbon Loudspeaker]] along with Dr. Gerwin Erlach in 1924<ref name="Ribbon">{{Cite news|url=http://www.hi-fiworld.co.uk/loudspeakers/66-knowledge/152-historically-speaking-part-ii.html|title=Historically Speaking|publisher=Hifi World|date=April 2008|accessdate=April 2012}}</ref>  and later made many significant contributions in the areas of semiconductor devices, technical physics and technology.


[[File:Emmy noether postcard 1915.jpg|thumb|A 1915 postcard from one of the pioneers of commutative algebra, [[Emmy Noether]], to E. Fischer, discussing her work in commutative algebra.]]
==Education==
He graduated from the [[Steglitz Gymnasium]], Berlin, Germany in 1904. He obtained his BS in [[Physics]], at the [[University of Berlin]] in 1908. He obtained his PhD in Physics at the [[University of Berlin]] in 1912 under [[Max Planck]] and [[Heinrich Rubens]], with a thesis entitled: ''Zur relativtheoretischen Energetik und Dynamik''.


'''Commutative algebra''' is the branch of [[algebra]] that studies [[commutative ring]]s, their [[ideal (ring theory)|ideals]], and [[module (mathematics)|modules]] over such rings. Both [[algebraic geometry]] and [[algebraic number theory]] build on commutative algebra. Prominent examples of commutative rings include [[polynomial ring]]s, rings of [[algebraic integer]]s, including the ordinary [[integer]]s <math>\mathbb{Z}</math>, and [[p-adic number|''p''-adic integer]]s.<ref>Atiyah and Macdonald, 1969, Chapter 1</ref>
==Career==
His postdoctoral period was spent at [[University of Jena]] (1912–14). He then lectured at the [[University of Würzburg]] (1919–23). He became Professor of Theoretical Physics, [[University of Rostock]] (1923–27). For two periods he worked at the Siemens Research laboratories (1914–19, 1927–58).


Commutative algebra is the main technical tool in the local study of [[scheme (mathematics)|schemes]].
==Inventions==
In 1924, Schottky co-invented the [[Ribbon microphone]] along with Dr. Gerwin Erlach. The idea was that a very fine ribbon suspended in a magnetic field could generate electric signals. This in turn lead also to the invention of the [[Loudspeaker#Ribbon_and_planar_magnetic_loudspeakers|Ribbon Loudspeaker]] by using it in the reverse order, but not until high flux permanent magnets became available in the late 1930s.<ref name="Ribbon"></ref>


The study of rings which are not necessarily commutative is known as [[noncommutative algebra]]; it includes [[ring theory]], [[representation theory]], and the theory of [[Banach algebra]]s.
==Major scientific achievements==
Possibly, in retrospect, Schottky's most important scientific achievement was to develop (in 1914) the well-known classical formula, now written -''q''<sup>2</sup>/16π''ε''<sub>0</sub>''x'', for the interaction energy between a point [[electric charge|charge]] ''q'' and a ''flat'' metal surface, when the charge is at a distance ''x'' from the surface. Owing to the method of its derivation, this interaction is called the "image potential energy" (image PE). Schottky based his work on earlier work by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] relating to the image PE for a sphere. Schottky's image PE has become a standard component in simple models of the barrier to motion, ''M''(''x''), experienced by an electron on approaching a metal surface or a metal–[[semiconductor]] interface from the inside. (This ''M''(''x'') is the quantity that appears when the one-dimensional, one-particle, [[Schrödinger equation]] is written in the form


==Overview==
:<math>\frac{d^2}{dx^2} \Psi(x) = \frac{2m}{\hbar^2} M(x) \Psi(x) .</math>
Commutative algebra is essentially the study of the rings occurring in [[algebraic number theory]] and [[algebraic geometry]]


In algebraic number theory, the rings of [[algebraic integer]]s are [[Dedekind ring]]s, which constitute therefore an important class of commutative rings. Considerations related to [[modular arithmetic]] have led to the notion of [[valuation ring]]. The restriction of [[algebraic field extension]]s to subrings has led to the notions of [[integral extension]]s and [[integrally closed domain]]s as well as the notion of [[ramification]] of an extension of valuation rings.
Here, <math> \hbar </math> is [[Planck's constant]] divided by 2π, and ''m'' is the [[electron mass]].)


The notion of [[localization of a ring]] (in particular the localization with respect to a [[prime ideal]], the localization consisting in inverting a single element and the [[total quotient ring]]) is one of the main differences between commutative algebra and the theory of non-commutative rings. It leads to an important class of commutative rings, the [[local ring]]s that have only one [[maximal ideal]]. The set of the prime ideals of a commutative ring is naturally equipped with a [[topological space|topology]], the [[Zariski topology]]. All these notions are widely used in algebraic geometry and are the basic technical tools for the definition of [[scheme theory]], a generalization of algebraic geometry introduced by [[Grothendieck]].
The image PE is usually combined with terms relating to an applied [[electric field]] ''F'' and to the height ''h'' (in the absence of any field) of the barrier. This leads to the following expression for the dependence of the barrier energy on distance ''x'', measured from the "electrical surface" of the metal, into the [[vacuum]] or into the [[semiconductor]]:


Many other notions of commutative algebra are counterparts of geometrical notions occurring in algebraic geometry. This is the case of [[Krull dimension]], [[primary decomposition]], [[regular ring]]s, [[Cohen–Macaulay ring]]s, [[Gorenstein ring]]s and many other notions.
:<math> M(x) = \; h -eFx - e^2/4 \pi \epsilon_0 \epsilon_r x \;. </math>
== History ==
The subject, first known as [[ideal theory]], began with [[Richard Dedekind]]'s work on [[Ideal (ring theory)|ideal]]s, itself based on the earlier work of [[Ernst Kummer]] and [[Leopold Kronecker]]. Later, [[David Hilbert]] introduced the term ''ring'' to generalize the earlier term ''number ring''. Hilbert introduced a more abstract approach to replace the more concrete and computationally oriented methods grounded in such things as [[complex analysis]] and classical [[invariant theory]].  In turn, Hilbert strongly influenced [[Emmy Noether]], who recast many earlier results in terms of an [[ascending chain condition]], now known as the Noetherian condition. Another important milestone was the work of Hilbert's student [[Emanuel Lasker]], who introduced [[primary ideal]]s and proved the first version of the [[Lasker–Noether theorem]].


The main figure responsible for the birth of commutative algebra as a mature subject was [[Wolfgang Krull]], who introduced the fundamental notions of [[Localization of a ring|localization]] and [[Completion (ring theory)|completion]] of a ring, as well as that of [[regular local ring]]s. He established the concept of the [[Krull dimension]] of a ring, first for [[Noetherian rings]] before moving on to expand his theory to cover general [[valuation ring]]s and [[Krull ring]]s. To this day, [[Krull's principal ideal theorem]] is widely considered the single most important foundational theorem in commutative algebra. These results paved the way for the introduction of commutative algebra into algebraic geometry, an idea which would revolutionize the latter subject.
Here, ''e'' is the [[elementary charge|elementary positive charge]], ''ε''<sub>0</sub> is the [[electric constant]] and ''ε''<sub>r</sub> is the [[relative permittivity]] of the second medium (=1 for [[vacuum]]). In the case of a [[metal–semiconductor junction]], this is called a [[Schottky barrier]]; in the case of the metal-vacuum interface, this is sometimes called a [[Field electron emission|Schottky–Nordheim barrier]]. In many contexts, ''h'' has to be taken equal to the local [[work function]] ''φ''.


Much of the modern development of commutative algebra emphasizes [[module (mathematics)|modules]].  Both  ideals of a ring ''R'' and ''R''-algebras are special cases of ''R''-modules, so module theory encompasses both ideal theory and the theory of [[ring extensions]]. Though it was already incipient in [[Kronecker|Kronecker's]] work, the modern approach to commutative algebra using module theory is usually credited to [[Wolfgang Krull|Krull]] and [[Emmy Noether|Noether]].
This [[Field electron emission|Schottky–Nordheim barrier]] (SN barrier) has played in important role in the theories of [[thermionic emission]] and of [[field electron emission]]. Applying the field causes lowering of the barrier, and thus enhances the emission current in [[thermionic emission]]. This is called the "[[Thermionic emission|Schottky effect]]", and the resulting emission regime is called "[[Thermionic emission|Schottky emission]]".


==Main tools and results==
In 1923 Schottky suggested (incorrectly) that the experimental phenomenon then called autoelectronic emission and now called [[field electron emission]] resulted when the barrier was pulled down to zero. In fact, the effect is due to [[quantum tunnelling|wave-mechanical tunneling]], as shown by Fowler and Nordheim in 1928. But the [[Field electron emission|SN barrier]] has now become the standard model for the tunneling barrier.


===Noetherian rings===
Later, in the context of [[semiconductor devices]], it was suggested that a similar barrier should exist at the junction of a metal and a semiconductor. Such barriers are now widely known as [[Schottky barrier]]s, and considerations apply to the transfer of electrons across them that are analogous to the older considerations of how electrons are emitted from a metal into vacuum. (Basically, several emission regimes exist, for different combinations of field and temperature. The different regimes are governed by different approximate formulae.)
{{Main|Noetherian ring}}
In [[mathematics]], more specifically in the area of [[Abstract algebra|modern algebra]] known as [[Ring (mathematics)|ring theory]], a '''Noetherian ring''', named after [[Emmy Noether]], is a ring in which every non-empty set of [[ideal (ring theory)|ideal]]s has a maximal element. Equivalently, a ring is Noetherian if it satisfies the [[ascending chain condition]] on ideals; that is, given any chain:


:<math>I_1\subseteq\cdots I_{k-1}\subseteq I_{k}\subseteq I_{k+1}\subseteq\cdots</math>
When the whole behaviour of such interfaces is examined, it is found that they can act (asymmetrically) as a special form of electronic diode, now called a [[Schottky diode]]. In this context, the metal–semiconductor junction is known as a "[[Schottky contact|Schottky (rectifying) contact']]".


there exists an ''n'' such that:
Schottky's contributions, in surface science/emission electronics and in semiconductor-device theory, now form a significant and pervasive part of the background to these subjects. It could possibly be argued that – perhaps because they are in the area of technical physics – they are not as generally well recognized as they ought to be.


:<math>I_{n}=I_{n+1}=\cdots</math>
==Awards==
He was awarded the [[Royal Society]]'s [[Hughes medal]] in 1936 for his discovery of the [[Schrot effect]] (spontaneous current variations in high-vacuum discharge tubes, called by him the "Schrot effect": literally, the "small shot effect") in [[thermionic emission]] and his invention of the screen-grid tetrode and a [[superheterodyne]] method of receiving wireless signals.


For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated. (The result is due to [[I. S. Cohen]].)
In 1964 he received the [[Werner von Siemens Ring]] honoring his ground-breaking work on the physical understanding of many phenomena that led to many important technical appliances, among them [[tube amplifier]]s and [[semiconductor]]s.


The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring. For instance, the ring of [[integer]]s and the [[polynomial ring]] over a [[Field (mathematics)|field]] are both Noetherian rings, and consequently, such theorems as the [[Lasker–Noether theorem]], the [[Krull intersection theorem]], and the [[Hilbert's basis theorem]] hold for them. Furthermore, if a ring is Noetherian, then it satisfies the [[descending chain condition]] on ''[[prime ideal]]s''. This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the [[Krull dimension]].
==Controversy==
The invention of superheterodyne is usually attributed to [[Edwin Armstrong]]. However, Schottky published an article in [[Proc. IRE]] that he had also invented something similar.
* 1939: first [[p-n junction]]


===Hilbert's basis theorem===
==Personal life==
{{Main|Hilbert's basis theorem}}
His father was [[mathematician]] [[Friedrich Hermann Schottky]] (1851–1935). His wife was Elizabeth and they had one daughter and two sons. His father was appointed professor of mathematics at the [[University of Zurich]] in 1882, and he was born 4 years later. The family then moved back to Germany in 1892, where his father took up an appointment at the [[University of Marburg]].
<blockquote>'''Theorem.''' If ''R'' is a left (resp. right) [[Noetherian ring]], then the [[polynomial ring]] ''R''[''X''] is also a left (resp. right) Noetherian ring.</blockquote>


Hilbert's basis theorem has some immediate corollaries:  
==Legacy==
[[:de:Walter Schottky Institut|Walter Schottky Institute]] (Germany) was named after him. The [[:de:Walter-Schottky-Preis|Walter H. Schottky prize]] is named after him.


#By induction we see that <math>R[X_0,\dotsc,X_{n-1}]</math> will also be Noetherian.
==Books written by Schottky==
#Since any [[affine variety]] over <math>R^n</math> (i.e. a locus-set of a collection of polynomials) may be written as the locus of an ideal <math>\mathfrak a\subset R[X_0, \dotsc, X_{n-1}]</math> and further as the locus of its generators, it follows that every affine variety is the locus of finitely many polynomials — i.e. the intersection of finitely many [[hypersurface]]s.
* ''Thermodynamik'', Julius Springer, Berlin, Germany, 1929.
#If <math>A</math> is a finitely-generated <math>R</math>-algebra, then we know that <math>A \simeq R[X_0, \dotsc, X_{n-1}] / \mathfrak a</math>, where <math>\mathfrak a</math> is an ideal. The basis theorem implies that <math>\mathfrak a</math> must be finitely generated, say <math>\mathfrak a = (p_0,\dotsc, p_{N-1})</math>, i.e. <math>A</math> is [[Glossary_of_ring_theory#Finitely_presented_algebra|finitely presented]].
* ''Physik der Glühelektroden'', Akademische Verlagsgesellschaft, Leipzig, 1928.


===Primary decomposition===
==See also==
{{Main|Primary decomposition}}
* [[Schottky defect]]
An ideal ''Q'' of a ring is said to be ''[[Primary ideal|primary]]'' if ''Q'' is [[proper subset|proper]] and whenever ''xy'' ∈ ''Q'', either ''x'' ∈ ''Q'' or ''y<sup>n</sup>'' ∈ ''Q'' for some positive integer ''n''. In '''Z''', the primary ideals are precisely the ideals of the form (''p<sup>e</sup>'') where ''p'' is prime and ''e'' is a positive integer. Thus, a primary decomposition of (''n'') corresponds to representing (''n'') as the intersection of finitely many primary ideals.


The ''[[Lasker–Noether theorem]]'', given here, may be seen as a certain generalization of the fundamental theorem of arithmetic:
==References==
{{Reflist}}


<blockquote>'''Lasker-Noether Theorem.''' Let ''R'' be a commutative Noetherian ring and let ''I'' be an ideal of ''R''. Then ''I'' may be written as the intersection of finitely many primary ideals with distinct [[Radical of an ideal|radicals]]; that is:
==External links==
* [http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_2/advanced/t2_1_3.html Walter Schottky]
* [http://www.webcitation.org/query?url=http://www.geocities.com/bioelectrochemistry/schottky.htm&date=2009-10-25+19:12:49 Biography of Walter H. Schottky]
* [http://www.wsi.tum.de/ Walter Schottky Institut]
* {{DNB portal|118759183|TYP=}}
* [http://www.gnt-verlag.de/de/?id=88 Reinhard W. Serchinger: Walter Schottky – Atomtheoretiker und Elektrotechniker.] Sein Leben und Werk bis ins Jahr 1941. Diepholz; Stuttgart; Berlin: GNT-Verlag, 2008.
* [http://www.nndb.com/people/438/000172919/ Schottky's nndb profile]
* [http://genealogy.math.ndsu.nodak.edu/id.php?id=55830 Schottky's math genealogy]


: <math>I=\bigcap_{i=1}^t Q_i</math>
{{Use dmy dates|date=December 2010}}


with ''Q<sub>i</sub>'' primary for all ''i'' and Rad(''Q<sub>i</sub>'') ≠ Rad(''Q<sub>j</sub>'') for ''i'' ≠ ''j''. Furthermore, if:
{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
| NAME              = Schottky, Walter H.
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION =
| DATE OF BIRTH    = 23 July 1886
| PLACE OF BIRTH    = [[Zürich]], [[Switzerland]]
| DATE OF DEATH    = 4 March 1976
| PLACE OF DEATH    = [[Pretzfeld]], [[West Germany]]
}}
{{DEFAULTSORT:Schottky, Walter H.}}
[[Category:Semiconductor physicists]]
[[Category:1886 births]]
[[Category:1976 deaths]]
[[Category:German electrical engineers]]
[[Category:German physicists]]
[[Category:Werner von Siemens Ring laureates]]


: <math>I=\bigcap_{i=1}^k P_i</math>
[[be-x-old:Вальтэр Шоткі]]
 
[[ca:Walter H. Schottky]]
is decomposition of ''I'' with Rad(''P<sub>i</sub>'') ≠ Rad(''P<sub>j</sub>'') for ''i'' ≠ ''j'', and both decompositions of ''I'' are ''irredundant'' (meaning that no proper subset of either {''Q''<sub>1</sub>, ..., ''Q<sub>t</sub>''} or {''P''<sub>1</sub>, ..., ''P<sub>k</sub>''} yields an intersection equal to ''I''), ''t'' = ''k'' and (after possibly renumbering the ''Q<sub>i</sub>'') Rad(''Q<sub>i</sub>'') = Rad(''P<sub>i</sub>'') for all ''i''.</blockquote>
[[cs:Walter Schottky]]
 
[[de:Walter Schottky]]
For any primary decomposition of ''I'', the set of all radicals, that is, the set {Rad(''Q''<sub>1</sub>), ..., Rad(''Q<sub>t</sub>'')} remains the same by the Lasker–Noether theorem. In fact, it turns out that (for a Noetherian ring) the set is precisely the [[associated prime|assassinator]] of the module ''R''/''I''; that is, the set of all [[annihilator (ring theory)|annihilators]] of ''R''/''I'' (viewed as a module over ''R'') that are prime.
[[et:Walter Schottky]]
 
[[es:Walter H. Schottky]]
===Localization===
[[fr:Walter Schottky]]
{{main|Localization (algebra)}}
[[id:Walter H. Schottky]]
 
[[it:Walter Schottky]]
The [[localization (algebra)|localization]] is a formal way to introduce the "denominators" to given a ring or a module. That is, it introduces a new ring/module out of an existing one so that it consists of [[algebraic fraction|fractions]]  
[[lt:Walter Schottky]]
:<math>\frac{m}{s}</math>.
[[mr:वॉल्टर शॉट्की]]
where the [[denominator]]s ''s'' range in a given subset ''S'' of ''R''. The basic example is the construction of the ring '''Q''' of rational numbers from the ring '''Z''' of rational integers.
[[nl:Walter Schottky]]
 
[[ja:ヴァルター・ショットキー]]
===Completion===
[[pl:Walter Schottky]]
{{main|Completion (ring theory)}}
[[pt:Walter Schottky]]
A [[completion (ring theory)|completion]] is any of several related [[functor]]s on [[ring (mathematics)|ring]]s and [[module (mathematics)|modules]] that result in complete [[topological ring]]s and modules. Completion is similar  to [[localization of a ring|localization]], and together they are among the most basic tools in analysing [[commutative ring]]s. Complete commutative rings have simpler structure than the general ones and [[Hensel's lemma]] applies to them.
[[ro:Walter Schottky]]
 
[[ru:Шоттки, Вальтер]]
===Zariski topology on prime ideals===
[[uk:Вальтер Шотткі]]
{{main|Zariski topology}}
[[zh:華特·蕭特基]]
The [[Zariski topology]] defines a [[topological space|topology]] on the [[spectrum of a ring]] (the set of prime ideals).<ref>{{cite book
| last1 = Dummit
| first1 = D. S.
| last2 = Foote
| first2 = R.
| title = Abstract Algebra
| publisher = Wiley
| pages = 71–72
| year = 2004
| edition = 3
| isbn = 9780471433347
}}</ref>  In this formulation, the Zariski-closed sets are taken to be the sets
 
:<math>V(I) = \{P \in \operatorname{Spec}\,(A) \mid I \subseteq P\}</math>
 
where ''A'' is a fixed commutative ring and ''I'' is an ideal.  This is defined in analogy with the classical Zariski topology, where closed sets in affine space are those defined by polynomial equations . To see the connection with the classical picture, note that for any set ''S'' of polynomials (over an algebraically closed field), it follows from [[Hilbert's Nullstellensatz]] that the points of ''V''(''S'') (in the old sense) are exactly the tuples (''a<sub>1</sub>'', ..., ''a<sub>n</sub>'') such that (''x<sub>1</sub>'' - ''a<sub>1</sub>'', ..., ''x<sub>n</sub>'' - ''a<sub>n</sub>'') contains ''S''; moreover, these are maximal ideals and by the "weak" Nullstellensatz, an ideal of any affine coordinate ring is maximal if and only if it is of this form.  Thus, ''V''(''S'') is "the same as" the maximal ideals containing ''S''.  Grothendieck's innovation in defining Spec was to replace maximal ideals with all prime ideals; in this formulation it is natural to simply generalize this observation to the definition of a closed set in the spectrum of a ring.
 
==Examples==
 
The fundamental example in commutative algebra is the ring of integers <math>\mathbb{Z}</math>. The existence of primes and
the unique factorization theorem laid the foundations for concepts such as [[Noetherian ring]]s and the [[primary decomposition]].
 
Other important examples are:
*[[Polynomial ring]]s <math>R[x_1,...,x_n]</math>
*The [[p-adic integer]]s
*Rings of [[algebraic integer]]s.
 
==Connections with algebraic geometry==
Commutative algebra (in the form of [[polynomial ring]]s and their quotients, used in the definition of [[algebraic varieties]]) has always been a part of [[algebraic geometry]]. However, in late 1950s, algebraic varieties were subsumed into [[Alexander Grothendieck]]'s concept of a [[scheme (mathematics)|scheme]]. Their local objects are affine schemes or prime spectra which are locally ringed spaces which form a category which is antiequivalent (dual) to the category of commutative unital rings, extending the [[duality (category theory)|duality]] between the category of affine algebraic varieties over a field ''k'', and the category of finitely generated reduced ''k''-algebras. The gluing is along Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes. The Zariski topology in the set theoretic sense is then replaced by a Zariski topology in the sense of [[Grothendieck topology]]. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and more sensitive examples than the crude Zariski topology, namely the [[étale topology]], and the two flat Grothendieck topologies: fppf and fpqc; nowadays some other examples became prominent including [[Nisnevich topology]]. Sheaves can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions leading to Artin stacks and, even finer, [[Deligne-Mumford stack]]s, both often called algebraic stacks.
 
== See also ==
 
* [[List of commutative algebra topics]]
* [[Glossary of commutative algebra]]
* [[Combinatorial commutative algebra]]
* [[Gröbner basis]]
* [[Homological algebra]]
 
== References ==
{{reflist}}
* [[Michael Atiyah]] & [[Ian G. Macdonald]], ''[[Introduction to Commutative Algebra]]'', Massachusetts : Addison-Wesley Publishing, 1969.
* [[Nicolas Bourbaki|Bourbaki, Nicolas]], ''Commutative algebra. Chapters 1--7''. Translated from the French. Reprint of the 1989 English translation. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1998. xxiv+625 pp. ISBN 3-540-64239-0
* [[Nicolas Bourbaki|Bourbaki, Nicolas]], ''Éléments de mathématique. Algèbre commutative. Chapitres 8 et 9''. (Elements of mathematics. Commutative algebra. Chapters 8 and 9) Reprint of the 1983 original. Springer, Berlin, 2006. ii+200 pp. ISBN 978-3-540-33942-7
* [[David Eisenbud]], ''[[Commutative Algebra With a View Toward Algebraic Geometry]]'', New York : Springer-Verlag, 1999.
* Rémi Goblot, "Algèbre commutative, cours et exercices corrigés", 2e édition, Dunod 2001, ISBN 2-10-005779-0
* Ernst Kunz, "Introduction to Commutative algebra and algebraic geometry", Birkhauser 1985,  ISBN 0-8176-3065-1
* Matsumura, Hideyuki, ''Commutative algebra''. Second edition. Mathematics Lecture Note Series, 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. xv+313 pp. ISBN 0-8053-7026-9
* Matsumura, Hideyuki, ''Commutative Ring Theory''. Second edition. Translated from the Japanese. Cambridge Studies in Advanced Mathematics, Cambridge, UK : Cambridge University Press, 1989. ISBN 0-521-36764-6
* [[Masayoshi Nagata|Nagata, Masayoshi]], ''Local rings''. Interscience Tracts in Pure and Applied Mathematics, No. 13. Interscience Publishers a division of John Wiley and Sons, New York-London 1962 xiii+234 pp.
* Miles Reid, ''[[Undergraduate Commutative Algebra]] (London Mathematical Society Student Texts)'', Cambridge, UK : Cambridge University Press, 1996.
* [[Jean-Pierre Serre]], ''Local algebra''. Translated from the French by CheeWhye Chin and revised by the author. (Original title: ''Algèbre locale, multiplicités'') Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2000. xiv+128 pp. ISBN 3-540-66641-9
* Sharp, R. Y., ''Steps in commutative algebra''. Second edition. London Mathematical Society Student Texts, 51. Cambridge University Press, Cambridge, 2000. xii+355 pp. ISBN 0-521-64623-5
* [[Oscar Zariski|Zariski, Oscar]]; [[Pierre Samuel|Samuel, Pierre]], ''Commutative algebra''. Vol. 1, 2. With the cooperation of I. S. Cohen. Corrected reprinting of the 1958, 1960 edition. Graduate Texts in Mathematics, No. 28, 29. Springer-Verlag, New York-Heidelberg-Berlin, 1975.
* {{Citation | last1=Zeidler | first1=A. Bernhard | title=Abstract Algebra | year=2014 | url=https://wuala.com/Chronicler/Mathematics/algebra.pdf }} A web-book on algebra and commutative algebra. Warning: work in progress! Free downloadable PDF under Open Publication License.
 
[[Category:Commutative algebra| ]]

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Template:Infobox scientist Walter Hermann Schottky (23 July 1886, Zürich, Switzerland – 4 March 1976, Pretzfeld, West Germany) was a German physicist who played a major early role in developing the theory of electron and ion emission phenomena, invented the screen-grid vacuum tube in 1915 and the pentodePotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. in 1919 while working at Siemens, co-invented the Ribbon microphone and Ribbon Loudspeaker along with Dr. Gerwin Erlach in 1924[1] and later made many significant contributions in the areas of semiconductor devices, technical physics and technology.

Education

He graduated from the Steglitz Gymnasium, Berlin, Germany in 1904. He obtained his BS in Physics, at the University of Berlin in 1908. He obtained his PhD in Physics at the University of Berlin in 1912 under Max Planck and Heinrich Rubens, with a thesis entitled: Zur relativtheoretischen Energetik und Dynamik.

Career

His postdoctoral period was spent at University of Jena (1912–14). He then lectured at the University of Würzburg (1919–23). He became Professor of Theoretical Physics, University of Rostock (1923–27). For two periods he worked at the Siemens Research laboratories (1914–19, 1927–58).

Inventions

In 1924, Schottky co-invented the Ribbon microphone along with Dr. Gerwin Erlach. The idea was that a very fine ribbon suspended in a magnetic field could generate electric signals. This in turn lead also to the invention of the Ribbon Loudspeaker by using it in the reverse order, but not until high flux permanent magnets became available in the late 1930s.[1]

Major scientific achievements

Possibly, in retrospect, Schottky's most important scientific achievement was to develop (in 1914) the well-known classical formula, now written -q2/16πε0x, for the interaction energy between a point charge q and a flat metal surface, when the charge is at a distance x from the surface. Owing to the method of its derivation, this interaction is called the "image potential energy" (image PE). Schottky based his work on earlier work by Lord Kelvin relating to the image PE for a sphere. Schottky's image PE has become a standard component in simple models of the barrier to motion, M(x), experienced by an electron on approaching a metal surface or a metal–semiconductor interface from the inside. (This M(x) is the quantity that appears when the one-dimensional, one-particle, Schrödinger equation is written in the form

d2dx2Ψ(x)=2mℏ2M(x)Ψ(x).

Here, ℏ is Planck's constant divided by 2π, and m is the electron mass.)

The image PE is usually combined with terms relating to an applied electric field F and to the height h (in the absence of any field) of the barrier. This leads to the following expression for the dependence of the barrier energy on distance x, measured from the "electrical surface" of the metal, into the vacuum or into the semiconductor:

M(x)=h−eFx−e2/4πϵ0ϵrx.

Here, e is the elementary positive charge, ε0 is the electric constant and εr is the relative permittivity of the second medium (=1 for vacuum). In the case of a metal–semiconductor junction, this is called a Schottky barrier; in the case of the metal-vacuum interface, this is sometimes called a Schottky–Nordheim barrier. In many contexts, h has to be taken equal to the local work function φ.

This Schottky–Nordheim barrier (SN barrier) has played in important role in the theories of thermionic emission and of field electron emission. Applying the field causes lowering of the barrier, and thus enhances the emission current in thermionic emission. This is called the "Schottky effect", and the resulting emission regime is called "Schottky emission".

In 1923 Schottky suggested (incorrectly) that the experimental phenomenon then called autoelectronic emission and now called field electron emission resulted when the barrier was pulled down to zero. In fact, the effect is due to wave-mechanical tunneling, as shown by Fowler and Nordheim in 1928. But the SN barrier has now become the standard model for the tunneling barrier.

Later, in the context of semiconductor devices, it was suggested that a similar barrier should exist at the junction of a metal and a semiconductor. Such barriers are now widely known as Schottky barriers, and considerations apply to the transfer of electrons across them that are analogous to the older considerations of how electrons are emitted from a metal into vacuum. (Basically, several emission regimes exist, for different combinations of field and temperature. The different regimes are governed by different approximate formulae.)

When the whole behaviour of such interfaces is examined, it is found that they can act (asymmetrically) as a special form of electronic diode, now called a Schottky diode. In this context, the metal–semiconductor junction is known as a "Schottky (rectifying) contact'".

Schottky's contributions, in surface science/emission electronics and in semiconductor-device theory, now form a significant and pervasive part of the background to these subjects. It could possibly be argued that – perhaps because they are in the area of technical physics – they are not as generally well recognized as they ought to be.

Awards

He was awarded the Royal Society's Hughes medal in 1936 for his discovery of the Schrot effect (spontaneous current variations in high-vacuum discharge tubes, called by him the "Schrot effect": literally, the "small shot effect") in thermionic emission and his invention of the screen-grid tetrode and a superheterodyne method of receiving wireless signals.

In 1964 he received the Werner von Siemens Ring honoring his ground-breaking work on the physical understanding of many phenomena that led to many important technical appliances, among them tube amplifiers and semiconductors.

Controversy

The invention of superheterodyne is usually attributed to Edwin Armstrong. However, Schottky published an article in Proc. IRE that he had also invented something similar.

Personal life

His father was mathematician Friedrich Hermann Schottky (1851–1935). His wife was Elizabeth and they had one daughter and two sons. His father was appointed professor of mathematics at the University of Zurich in 1882, and he was born 4 years later. The family then moved back to Germany in 1892, where his father took up an appointment at the University of Marburg.

Legacy

Walter Schottky Institute (Germany) was named after him. The Walter H. Schottky prize is named after him.

Books written by Schottky

  • Thermodynamik, Julius Springer, Berlin, Germany, 1929.
  • Physik der Glühelektroden, Akademische Verlagsgesellschaft, Leipzig, 1928.

See also

References

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External links

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  1. ↑ 1.0 1.1 Template:Cite news
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