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| In [[mathematics]], specifically [[commutative algebra]], '''Hilbert's basis theorem''' states that every [[Ideal (ring theory)|ideal]] in the [[polynomial ring|ring of multivariate polynomials]] over a [[Noetherian ring]] is [[finitely generated module|finitely generated]]. This can be translated into [[algebraic geometry]] as follows: every [[algebraic set]] over a field can be described as the set of common roots of finitely many polynomial equations. {{harvs|txt|authorlink=David Hilbert|last=Hilbert|year=1890}} proved the theorem (for the special case of polynomial rings over a field) in the course of his proof of finite generation of rings of invariants.
| | If you have to accelerate the PC then we have come to the appropriate destination. I will show we, today, five quick ways which you can dramatically enhance your computer's performance.<br><br>However registry is conveniently corrupted plus damaged when you are utilizing the computer. Overtime, without right repair, it is loaded with mistakes and incorrect or even lost info which may create your program unable to work correctly or implement a certain task. And when a program cannot find the correct info, it will likely not learn what to do. Then it freezes up! That is the real cause of your trouble.<br><br>So what should you look for whenever you compare registry products. Many of the registry products available today, have especially similar attributes. The primary ones that you need to be trying to find are these.<br><br>First, constantly clean the PC and keep it without dust and dirt. Dirt clogs up all fans plus will cause the PC to overheat. You have to clean up disk area inside purchase to make your computer run quicker. Delete temporary plus unwanted files plus unused programs. Empty the recycle bin plus remove programs you may be not using.<br><br>After that, I also purchased the Regtool [http://bestregistrycleanerfix.com/registry-reviver registry reviver] Software, plus it further protected my laptop having system crashes. All my registry issues are fixed, plus I will work peacefully.<br><br>Files with all the DOC extension are furthermore susceptible to viruses, however this is solved by good antivirus programs. Another problem is that .doc files may be corrupted, unreadable or damaged due to spyware, adware, plus malware. These instances may prevent users from correctly opening DOC files. This is whenever effective registry cleaners become valuable.<br><br>Across the top of the scan results display page we see the tabs... Registry, Junk Files, Privacy, Bad Active X, Performance, etc. Every of these tabs can show you the results of that area. The Junk Files are mainly temporary files such as internet data, images, internet pages... And they are simply taking up storage.<br><br>Many people make the mistake of striving to fix Windows registry by hand. I strongly suggest you don't do it. Unless you're a computer expert, I bet you'll spend hours and hours learning the registry itself, let alone fixing it. And why if you waste a precious time inside understanding plus fixing something we know nothing about? Why not allow a smart and specialist registry cleaner do it for we? These software programs could do the job inside a far better technique! Registry cleaners are quite affordable because well; you pay a 1 time fee and use it forever. Also, most specialist registry cleaners are very reliable and user friendly. If you need more information on how to fix Windows registry, merely visit my website by clicking the link below! |
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| Hilbert produced an innovative proof by contradiction using [[mathematical induction]]; his method does not give an [[algorithm]] to produce the finitely many basis polynomials for a given ideal: it only shows that they must exist. One can determine basis polynomials using the method of [[Gröbner basis|Gröbner bases]].
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| ==Proof==
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| <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>
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| ''Remark.'' We will give two proofs, in both only the "left" case is considered, the proof for the right case is similar.
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| <br><br /> | |
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| '''First Proof.''' Suppose '''a''' ⊆ ''R''[''X''] were a non-finitely generated left-ideal. Then by recursion (using the [[axiom of countable choice]]) there is a sequence (''f<sub>n</sub>'')<sub>''n''∈'''N</sub>''' of polynomials such that if '''b'''<sub>''n''</sub> is the left ideal generated by ''f''<sub>0</sub>, ..., ''f''<sub>''n''−1</sub> then ''f<sub>n</sub>'' in '''a'''\'''b'''<sub>''n''</sub> is of minimal degree. It is clear that (deg(''f<sub>n</sub>''))<sub>''n''∈'''N</sub>''' is a non-decreasing sequence of naturals. Let ''a<sub>n</sub>'' be the leading coefficient of ''f<sub>n</sub>'' and let '''b''' be the left ideal in ''R'' generated by {''a''<sub>0</sub>, ''a''<sub>1</sub>, ...}. Since ''R'' is left-Noetherian, we have that '''b''' must be finitely generated; and since the ''a<sub>n</sub>'' comprise an ''R''-basis, it follows that for a finite amount of them, say {''a<sub>i</sub>'' : ''i'' < ''N''}, will suffice. So for example, <math>a_N=\sum_{i<N}u_{i}a_{i}\,</math> some ''u<sub>i</sub>'' in ''R''. Now consider
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| :<math>g \triangleq\sum_{i<N}u_{i}X^{\deg(f_{N})-\deg(f_{i})}f_{i},\,</math>
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| whose leading term is equal to that of ''f<sub>N</sub>''; moreover, ''g'' ∈ '''b'''<sub>''N''</sub>. However, ''f<sub>N</sub>'' ∉ '''b'''<sub>''N''</sub>, which means that ''f<sub>N</sub>''−''g'' ∈ '''a'''\'''b'''<sub>''N''</sub> has degree less than ''f<sub>N</sub>'', contradicting the minimality.
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| <br><br /> | |
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| '''Second Proof.''' Let '''a''' ⊆ ''R''[''X''] be a left-ideal. Let '''b''' be the set of leading coefficients of members of '''a'''. This is obviously a left-ideal over ''R'', and so is finitely generated by the leading coefficients of finitely many members of '''a'''; say ''f''<sub>0</sub>, ..., ''f''<sub>''N''−1</sub>. Let <math>d\triangleq\max_{i}\deg(f_{i}).\,</math> Let '''b'''<sub>''k''</sub> be the set of leading coefficients of members of '''a''', whose degree is ≤ ''k''. As before, the '''b'''<sub>''k''</sub> are left-ideals over ''R'', and so are finitely generated by the leading coefficients of finitely many members of '''a''', say <math>f^{(k)}_{0},\ldots,f^{(k)}_{N^{(k)}-1},\,</math> with degrees ≤ ''k''. Now let '''a'''* ⊆ ''R''[''X''] be the left-ideal generated by
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| :<math>\{f_{i},f^{(k)}_{j}:i<N,j<N^{(k)},k<d\}.\,</math>
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| We have '''a'''* ⊆ '''a''' and claim also '''a''' ⊆ '''a'''*. Suppose for the sake of contradiction this is not so. Then let ''h'' ∈ '''a'''\'''a'''* be of minimal degree, and denote its leading coefficient by ''a''.
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| :'''''Case 1:''''' deg(''h'') ≥ ''d''. Regardless of this condition, we have ''a'' ∈ '''b''', so is a left-linear combination <math>a=\sum_j u_j a_j\,</math> of the coefficients of the ''f<sub>j</sub>'' Consider <math>\tilde{h}\triangleq\sum_{j}u_{j}X^{\deg(h)-\deg(f_{j})}f_{j},\,</math> which has the same leading term as ''h''; moreover <math>\tilde{h}\in\mathfrak{a}^{\ast}\not\ni h\,</math> so <math>h-\tilde{h}\in\mathfrak{a}\setminus \mathfrak{a}^{\ast}\,</math> of degree < deg(''h''), contradicting minimality.
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| :'''''Case 2:''''' deg(''h'') = ''k'' < ''d''. Then ''a'' ∈ '''b'''<sub>''k''</sub> so is a left-linear combination <math>a=\sum_j u_j a^{(k)}_j</math> of the leading coefficients of the <math>f^{(k)}_j.</math> Considering <math>\tilde{h}\triangleq\sum_j u_j X^{\deg(h)-\deg(f^{(k)}_{j})}f^{(k)}_{j},</math> we yield a similar contradiction as in ''Case 1''.
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| Thus our claim holds, and '''a''' = '''a'''* which is finitely generated.
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| Note that the only reason we had to split into two cases was to ensure that the powers of ''X'' multiplying the factors, were non-negative in the constructions.
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| == Applications ==
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| Let ''R'' be a Noetherian commutative ring. Hilbert's basis theorem has some immediate corollaries. '''First,''' by induction we see that <math>R[X_{0},X_{1},\ldots,X_{n-1}]\,</math> will also be Noetherian. '''Second,''' since any [[affine variety]] over ''R<sup>n</sup>'' (''i.e.'' a locus-set of a collection of polynomials) may be written as the locus of an ideal <math>\mathfrak{a}\subseteq R[X_{0},X_{1},\ldots,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. '''Finally,''' if <math>\mathcal{A}\,</math> is a finitely-generated ''R''-algebra, then we know that <math>\mathcal{A}\cong R[X_{0},X_{1},\ldots,X_{n-1}]/\langle\mathfrak{a}\rangle\,</math> (''i.e.'' mod-ing out by relations), where '''a''' a set of polynomials. We can assume that '''a''' is an ideal and thus is finitely generated. So <math>\mathcal{A}\,</math> is a free ''R''-algebra (on ''n'' generators) generated by finitely many relations <math>\mathcal{A}\cong R[X_{0},X_{1},\ldots,X_{n-1}]/\langle p_{0},\ldots,p_{N-1}\rangle</math>.
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| ==Mizar System==
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| The [[Mizar system|Mizar project]] has completely formalized and automatically checked a proof of Hilbert's basis theorem in the [http://www.mizar.org/JFM/Vol12/hilbasis.html HILBASIS file].
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| ==References==
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| * Cox, Little, and O'Shea, ''Ideals, Varieties, and Algorithms'', Springer-Verlag, 1997.
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| *{{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | title=Ueber die Theorie der algebraischen Formen | doi=10.1007/BF01208503 | year=1890 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=36 | issue=4 | pages=473–534}}
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| [[Category:Commutative algebra]]
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| [[Category:Invariant theory]]
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| [[Category:Articles containing proofs]]
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| [[Category:Theorems in abstract algebra]]
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If you have to accelerate the PC then we have come to the appropriate destination. I will show we, today, five quick ways which you can dramatically enhance your computer's performance.
However registry is conveniently corrupted plus damaged when you are utilizing the computer. Overtime, without right repair, it is loaded with mistakes and incorrect or even lost info which may create your program unable to work correctly or implement a certain task. And when a program cannot find the correct info, it will likely not learn what to do. Then it freezes up! That is the real cause of your trouble.
So what should you look for whenever you compare registry products. Many of the registry products available today, have especially similar attributes. The primary ones that you need to be trying to find are these.
First, constantly clean the PC and keep it without dust and dirt. Dirt clogs up all fans plus will cause the PC to overheat. You have to clean up disk area inside purchase to make your computer run quicker. Delete temporary plus unwanted files plus unused programs. Empty the recycle bin plus remove programs you may be not using.
After that, I also purchased the Regtool registry reviver Software, plus it further protected my laptop having system crashes. All my registry issues are fixed, plus I will work peacefully.
Files with all the DOC extension are furthermore susceptible to viruses, however this is solved by good antivirus programs. Another problem is that .doc files may be corrupted, unreadable or damaged due to spyware, adware, plus malware. These instances may prevent users from correctly opening DOC files. This is whenever effective registry cleaners become valuable.
Across the top of the scan results display page we see the tabs... Registry, Junk Files, Privacy, Bad Active X, Performance, etc. Every of these tabs can show you the results of that area. The Junk Files are mainly temporary files such as internet data, images, internet pages... And they are simply taking up storage.
Many people make the mistake of striving to fix Windows registry by hand. I strongly suggest you don't do it. Unless you're a computer expert, I bet you'll spend hours and hours learning the registry itself, let alone fixing it. And why if you waste a precious time inside understanding plus fixing something we know nothing about? Why not allow a smart and specialist registry cleaner do it for we? These software programs could do the job inside a far better technique! Registry cleaners are quite affordable because well; you pay a 1 time fee and use it forever. Also, most specialist registry cleaners are very reliable and user friendly. If you need more information on how to fix Windows registry, merely visit my website by clicking the link below!