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| | Getting a superb night's sleep isn't only reserved for kids. As we get older, our bodies need longer to rest and recuperate each work day. With many people reporting more difficulty sleeping as they get older, those precious hours spent in the sack are answer to positive your body gets the downtime it. As long as a lot of them relaxing!<br><br><br><br>A. Strength training circuits: Perform 5-10 multi-jointed moves (i.e. squat and push up) using body weight, stability balls, and/or dumbbells. When completed within a circuit simply no rest between each move a pretty mean cardio workout will result.<br><br>Green Tea is a fabulous antioxidant. To produce your cells from wreck. Green Tea is also wonderful in cancer treatment and cancer prevention. There are some benefits to getting Green Tea besides fat loss.<br><br>All because of these factors causes an increased amount of body entire body fat. It's lovely staying thin for cosmetic reasons and pride. But really, we would like to have low body fat and more muscle aggregate. High body fat increases our risk of cancer, cardiovascular disease, arthritis and depression symptoms. So many serious medical conditions are deteriorated by developing a higher number of body excessive fat. That should be our motivation for being leaner and thinner.<br><br>Crash dieting (extreme caloric restriction) rarely works. All it does is shut your metabolism down a short time after setting up your diet, and/or catabolizes a involving muscle. Shoot for a 400-600 calorie a [http://forskolinbellybuster.net/ Forskolin Belly Buster] day restriction, and along with reasonable cardio you will establish a decent deficit. One does are already doing the same cardio when can or care to, you may need a bigger calorie decrease. This all varies a LOT between people of different bodyweights and the entire body fat compositions as well as folks metabolism. So while the suggested number will are suitable for many people, it might for every one of.<br><br>A 1980`s study revealed the effectiveness of a transdermal yohimbe, [http://forskolinbellybuster.net/ Forskolin Belly Buster], and aminophylline concoction. Everyone felt this was a miracle in the development of weight loss but didn`t consider the negatives requirements brought. People who didn`t know anything to your production of supplements would [http://forskolinbellybuster.net/ Forskolin Belly Buster] Review put together a cheap formula and tell everyone about review ? their product was, individuals off of bunk pieces. Another problem is that the particular good companies who can get the formula right still troubles generating them work effectively. The transdermal industry is still being perfected, but there are nevertheless methods the place can be familiar with help while using the weight loss process.<br><br>The thermogenic fat burners are without ephedra. People with heart conditions or other medical conditions should not take ephedra, so thermogenic diet supplements might use you. Ephedra can also cause of which you feel jittery, almost like you're on caffeine considerable. Other fat burners go about doing contain caffeine to increase you fat burning. It's known that dietary supplements without ephedra don't work quite also as people. The main point is increase your as well as that will burn more calories each day, plus will happen while you're resting. Stimulant-free themogenics can be found. These have the freedom of even caffeine. not lose as much right away, but they may be safer alternative. |
| In [[ring theory]], a branch of [[abstract algebra]], a '''commutative ring''' is a [[Ring (mathematics)|ring]] in which the multiplication operation is [[commutative]]. The study of commutative rings is called [[commutative algebra]].
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| Some specific kinds of commutative rings are given with the following chain of [[subclass (set theory)|class inclusions]]:
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| : '''Commutative rings''' ⊃ '''[[integral domain]]s''' ⊃ '''[[integrally closed domain]]s''' ⊃ '''[[unique factorization domain]]s''' ⊃ '''[[principal ideal domain]]s''' ⊃ '''[[Euclidean domain]]s''' ⊃ '''[[field (mathematics)|field]]s'''
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| == Definition and first examples ==
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| ===Definition===
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| {{Details|Ring (mathematics)|the definition of rings|Ring}}
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| A ''ring'' is a [[Set (mathematics)|set]] ''R'' equipped with two [[binary operation]]s, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "⋅"; e.g. ''a'' + ''b'' and ''a'' ⋅ ''b''. To form a ring these two operations have to satisfy a number of properties: the ring has to be an [[abelian group]] under addition as well as a [[monoid]] under multiplication, where multiplication [[distributive law|distributes]] over addition; i.e., ''a'' ⋅ (''b'' + ''c'') = (''a'' ⋅ ''b'') + (''a'' ⋅ ''c''). The identity elements for addition and multiplication are denoted 0 and 1, respectively.
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| If, as well, the multiplication is also commutative:
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| :''a'' ⋅ ''b'' = ''b'' ⋅ ''a''
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| then the ring ''R'' is called ''commutative''. In the remainder of this article, all rings will be commutative, unless explicitly stated otherwise.
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| ===First examples===
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| An important example, and in some sense crucial, is the [[integer|ring of integer]]s '''Z''' with the two operations of addition and multiplication. As the multiplication of integers is a commutative operation, this is a commutative ring. It is usually denoted '''Z''' as an abbreviation of the [[German language|German]] word ''Zahlen'' (numbers).
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| A [[field (mathematics)|field]] is a commutative ring where every [[0 (number)|non-zero]] element ''a'' is invertible; i.e., has a multiplicative inverse ''b'' such that ''a'' ⋅ ''b'' = 1. Therefore, by definition, any field is a commutative ring. The [[rational number|rational]], [[real number|real]] and [[complex number|complex]] numbers form fields.
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| The ring of 2×2 [[Matrix (mathematics)|matrices]] is ''not'' commutative, since [[matrix multiplication]] fails to be commutative, as the following example shows:
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| :<math>\begin{align}
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| \begin{bmatrix}
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| 1 & 1\\
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| 0 & 1\\
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| \end{bmatrix}\cdot
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| \begin{bmatrix}
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| 1 & 1\\
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| 1 & 0\\
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| \end{bmatrix} &=
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| \begin{bmatrix}
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| 2 & 1\\
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| 1 & 0\\
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| \end{bmatrix}\\
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| \begin{bmatrix}
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| 1 & 1\\
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| 1 & 0\\
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| \end{bmatrix}\cdot
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| \begin{bmatrix}
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| 1 & 1\\
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| 0 & 1\\
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| \end{bmatrix} &=
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| \begin{bmatrix}
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| 1 & 2\\
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| 1 & 1\\
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| \end{bmatrix}
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| \end{align}</math>
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| However, matrices that can be [[diagonalizable matrix|diagonalized]] with the same [[Matrix similarity|similarity transformation]] do form a commutative ring. An example is the set of matrices of [[divided difference#Matrix form|divided differences]] with respect to a fixed set of nodes.
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| If ''R'' is a given commutative ring, then the set of all [[polynomial]]s in the variable ''X'' whose coefficients are in ''R'' forms the [[polynomial ring]], denoted ''R''[''X'']. The same holds true for several variables.
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| If ''V'' is some [[topological space]], for example a subset of some '''R'''<sup>''n''</sup>, real- or complex-valued [[continuous function]]s on ''V'' form a commutative ring. The same is true for [[differentiable function|differentiable]] or [[holomorphic function]]s, when the two concepts are defined, such as for ''V'' a [[complex manifold]].
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| == Ideals and the spectrum ==
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| {{hatnote|In the following, R denotes a commutative ring.}}
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| In contrast to fields, where every nonzero element is multiplicatively invertible, the theory of rings is more complicated. There are several notions to cope with that situation. First, an element ''a'' of ring ''R'' is called a [[unit (algebra)|unit]] if it possesses a multiplicative inverse. Another particular type of element is the [[zero divisor]]s, i.e. a non-zero element ''a'' such that there exists a non-zero element ''b'' of the ring such that ''ab = 0''. If ''R'' possesses no zero divisors, it is called an [[integral domain]] since it closely resembles the integers in some ways.
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| Many of the following notions also exist for not necessarily commutative rings, but the definitions and properties are usually more complicated. For example, all ideals in a commutative ring are automatically [[two-sided ideal|two-sided]], which simplifies the situation considerably.
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| ===Ideals and factor rings===
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| {{Main|Ideal (ring theory)|l1=Ideal|Factor ring}}
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| The inner structure of a commutative ring is determined by considering its ideals, i.e. [[nonempty]] [[subset]]s that are closed under multiplication with arbitrary ring elements and addition: for all ''r'' in ''R'', ''i'' and ''j'' in ''I'', both ''ri'' and ''i'' + ''j'' are required to be in ''I''. Given any subset ''F'' = {''f''<sub>''j''</sub>}<sub>''j'' ∈ ''J''</sub> of ''R'' (where ''J'' is some index set), the ideal ''generated by F'' is the smallest ideal that contains ''F''. Equivalently, it is given by finite [[linear combination]]s
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| :''r''<sub>1</sub>''f''<sub>1</sub> + ''r''<sub>2</sub>''f''<sub>2</sub> + ... + ''r''<sub>''n''</sub>''f''<sub>''n''</sub>.
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| An ideal generated by one element is called [[principal ideal]]. A ring all of whose ideals are principal is called a [[principal ideal ring]], two important cases are '''Z''' and ''k''[''X''], the polynomial ring over a field ''k''. Any ring has two ideals, namely the [[0 (number)|zero ideal]] {0} and ''R'', the whole ring. Any ideal that is not contained in any proper ideal (i.e. ≠''R'') is called [[maximal ideal|maximal]]. <cite id=characterisaion_of_maximal_ideals>An ideal ''m'' is maximal [[if and only if]] ''R'' / ''m'' is a field.</cite> <cite id=existence_of_maximal_ideals>Any ring possesses at least one maximal ideal, a statement following from [[Zorn's lemma]], which is equivalent to the [[axiom of choice]].</cite>
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| The definition of ideals is such that "dividing" ''I'' "out" gives another ring, the ''factor ring'' ''R'' / ''I'': it is the set of [[coset]]s of ''I'' together with the operations
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| :(''a'' + ''I'') + (''b'' + ''I'') = (''a'' + ''b'') + I and (''a'' + ''I'')(''b'' + ''I'') = ''ab'' + ''I''.
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| For example, the ring '''Z'''/''n'''''Z''' (also denoted '''Z'''<sub>''n''</sub>), where ''n'' is an integer, is the ring of integers modulo ''n''. It is the basis of [[modular arithmetic]].
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| === Localizations===
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| {{Main|Localization of a ring}}
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| The ''localization'' of a ring is the counterpart to factor rings insofar as in a factor ring ''R'' / ''I'' certain elements (namely the elements of ''I'') become zero, whereas in the localization certain elements are rendered invertible, i.e. multiplicative inverses are added to the ring. Concretely, if ''S'' is a [[multiplicatively closed subset]] of ''R'' (i.e. whenever ''s'', ''t'' ∈ ''S'' then so is ''st'') then the ''localization'' of ''R'' at ''S'', or ''ring of fractions'' with denominators in ''S'', usually denoted ''S''<sup>−1</sup>''R'' consists of symbols
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| :<math>\frac{r}{s}</math> with ''r'' ∈ ''R'', ''s'' ∈ ''S''
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| subject to certain rules that mimick the cancellation familiar from rational numbers. Indeed, in this language '''Q''' is the localization of '''Z''' at all nonzero integers. This construction works for any integral domain ''R'' instead of '''Z'''. The localization (''R'' \ {0})<sup>−1</sup>''R'' is called the [[quotient field]] of ''R''. If ''S'' consists of the powers of one fixed element ''f'', the localisation is written ''R''<sub>''f''</sub>.
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| ===Prime ideals and the spectrum===
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| {{Main|Prime ideal|Spectrum of a ring}}
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| A particularly important type of ideals is ''prime ideals'', often denoted ''p''. This notion arose when algebraists (in the 19th century) realized that, unlike in '''Z''', in many rings there is no [[fundamental theorem of arithmetic|unique factorization into prime numbers]]. (Rings where it does hold are called [[unique factorization domain]]s.) By definition, a prime ideal is a proper ideal such that, whenever the product ''ab'' of any two ring elements ''a'' and ''b'' is in ''p'', at least one of the two elements is already in ''p''. (The opposite conclusion holds for any ideal, by definition). Equivalently, the factor ring ''R'' / ''p'' is an integral domain. Yet another way of expressing the same is to say that the [[Complement (set theory)|complement]] ''R'' \ ''p'' is multiplicatively closed. The localisation (''R'' \ ''p'')<sup>−1</sup>''R'' is important enough to have its own notation: ''R''<sub>''p''</sub>. This ring has only one maximal ideal, namely ''pR''<sub>''p''</sub>. Such rings are called [[local ring|local]].
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| By the [[#characterisaion of maximal ideals|above]], any maximal ideal is prime. Proving that an ideal is prime, or equivalently that a ring has no zero-divisors can be very difficult.
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| [[Image:Spec Z.png|right|400px|thumb|The spectrum of '''Z'''.]]
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| Prime ideals are the key step in interpreting a ring ''geometrically'', via the ''spectrum of a ring'' ''Spec R'': it is the set of all prime ideals of ''R''.<ref group=nb>This notion can be related to the [[Spectrum of an operator|spectrum]] of a linear operator, see [[Spectrum of a C*-algebra]] and [[Gelfand representation]].</ref> As noted [[#existence of maximal ideals|above]], there is at least one prime ideal, therefore the spectrum is nonempty. If ''R'' is a field, the only prime ideal is the zero ideal, therefore the spectrum is just one point. The spectrum of '''Z''', however, contains one point for the zero ideal, and a point for any prime number ''p'' (which generates the prime ideal ''p'''''Z'''). The spectrum is endowed with a topology called the [[Zariski topology]], which is determined by specifying that subsets ''D''(''f'') = {''p'' ∈ ''Spec R'', ''f'' ∉ ''p''}, where ''f'' is any ring element, be open. This topology tends to be different from those encountered in [[analysis]] or [[differential geometry]]; for example, there will generally be points which are not closed. The [[Closure (topology)|closure]] of the [[generic point|point corresponding to the zero ideal]] 0 ⊂ '''Z''', for example, is the whole spectrum of '''Z'''.
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| The notion of a spectrum is the common basis of commutative algebra and [[algebraic geometry]]. Algebraic geometry proceeds by endowing ''Spec R'' with a [[sheaf (mathematics)|sheaf]] <math>\mathcal O</math> (an entity that collects functions defined locally, i.e. on varying open subsets). The datum of the space and the sheaf is called an [[affine scheme]]. Given an affine scheme, the underlying ring ''R'' can be recovered as the [[global section]]s of <math>\mathcal O</math>. Moreover, the established one-to-one correspondence between rings and affine schemes is also compatible with ring homomorphisms: any ''f'' : ''R'' → ''S'' gives rise to a [[continuous map]] in the opposite direction
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| :''Spec S'' → ''Spec R'', ''q'' ↦ ''f''<sup>−1</sup>(''q''), i.e. any prime ideal of ''S'' is mapped to its [[preimage]] under ''f'', which is a prime ideal of ''R''.
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| The spectrum also makes precise the intuition that localisation and factor rings are complementary: the natural maps ''R'' → ''R''<sub>''f''</sub> and ''R'' → ''R'' / ''fR'' correspond, after endowing the spectra of the rings in question with their Zariski topology, to complementary [[open immersion|open]] and [[closed immersion]]s respectively.
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| Altogether the [[equivalence of categories|equivalence]] of the two said categories is very apt to reflect algebraic properties of rings in a geometrical manner. Affine schemes are–much the same way as [[manifold (mathematics)|manifolds]] are locally given by open subsets of '''R'''<sup>''n''</sup>–local models for [[scheme (mathematics)|schemes]], which are the object of study in algebraic geometry. Therefore, many notions that apply to rings and homomorphisms stem from geometric intuition.
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| ==Ring homomorphisms==
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| {{Main|Ring homomorphism}}
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| As usual in algebra, a function ''f'' between two objects that respects the structures of the objects in question is called [[homomorphism]]. In the case of rings, a ''ring homomorphism'' is a map ''f'' : ''R'' → ''S'' such that
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| :''f''(''a'' + ''b'') = ''f''(''a'') + ''f''(''b''), ''f''(''ab'') = ''f''(''a'')''f''(''b'') and ''f''(1) = 1.
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| These conditions ensure ''f''(0) = 0, but the requirement that the multiplicative identity element 1 is preserved under ''f'' would not follow from the two remaining properties. In such a situation ''S'' is also called an ''R''-algebra, by understanding that ''s'' in ''S'' may be multiplied by some ''r'' of ''R'', by setting
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| :''r'' · ''s'' := ''f''(''r'') · ''s''. | |
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| The ''kernel'' and ''image'' of ''f'' are defined by ker (''f'') = {''r'' ∈ ''R'', ''f''(''r'') = 0} and im (''f'') = ''f''(''R'') = {''f''(''r''), ''r'' ∈ ''R''}. The kernel is an [[ring ideal|ideal]] of ''R'', and the image is a [[subring]] of ''S''.
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| ==Modules==
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| {{Main|Module (mathematics)|l1=Modules}}
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| The outer structure of a commutative ring is determined by considering [[linear algebra]] over that ring, i.e., by investigating the theory of its [[module (mathematics)|modules]], which are similar to [[vector space]]s, except that the base is not necessarily a field, but can be any ring ''R''. The theory of ''R''-modules is significantly more difficult than linear algebra of vector spaces. Module theory has to grapple with difficulties such as modules not having bases, that the [[rank of a free module]] (i.e. the analog of the dimension of vector spaces) may not be well-defined and that submodules of finitely generated modules need not be finitely generated (unless ''R'' is Noetherian, see [[#submodules of f g modules|below]]).
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| Ideals within a ring ''R'' can be characterized as ''R''-modules which are submodules of ''R''. On the one hand, a good understanding of ''R''-modules necessitates enough information about ''R''. Vice versa, however, many techniques in commutative algebra that study the structure of ''R'', by examining its ideals, proceed by studying modules in general.
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| ==Noetherian rings==
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| {{Main|Noetherian ring}}
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| A ring is called ''Noetherian'' (in honor of [[Emmy Noether]], who developed this concept) if every [[ascending chain condition|ascending chain of ideals]]
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| :0 ⊆ ''I''<sub>0</sub> ⊆ ''I''<sub>1</sub> ... ⊆ ''I''<sub>''n''</sub> ⊆ ''I''<sub>''n'' + 1</sub> ⊆ ...
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| becomes stationary, i.e. becomes constant beyond some index ''n''. Equivalently, any ideal is generated by finitely many elements, or, yet equivalent, <cite id=submodules_of_f_g_modules>[[submodule]]s of finitely generated modules are finitely generated</cite>. A ring is called [[Artinian ring|Artinian]] (after [[Emil Artin]]), if every descending chain of ideals
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| :''R'' ⊇ ''I''<sub>0</sub> ⊇ ''I''<sub>1</sub> ... ⊇ ''I''<sub>''n''</sub> ⊇ ''I''<sub>''n'' + 1</sub> ⊇ ...
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| becomes stationary eventually. Despite the two conditions appearing symmetric, Noetherian rings are much more general than Artinian rings. For example, '''Z''' is Noetherian, since every ideal can be generated by one element, but is not Artinian, as the chain
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| :'''Z''' ⊋ 2'''Z''' ⊋ 4'''Z''' ⊋ 8'''Z''' ⊋ ...
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| shows. In fact, by the [[Hopkins–Levitzki theorem]], every Artinian ring is Noetherian.
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| Being Noetherian is an extremely important finiteness condition. The condition is preserved under many operations that occur frequently in geometry: if ''R'' is Noetherian, then so is the polynomial ring {{nowrap|''R''[''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub>]}} (by [[Hilbert's basis theorem]]), any localization ''S''<sup>−1</sup>''R'', factor rings ''R'' / ''I''.
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| ==Dimension==
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| {{Main|Krull dimension}}
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| The ''Krull dimension'' (or simply dimension) ''dim R'' of a ring ''R'' is a notion to measure the "size" of a ring, very roughly by the counting independent elements in ''R''. Precisely, it is defined as the supremum of lengths ''n'' of chains of prime ideals
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| :<math>\mathfrak{p}_0\subsetneq \mathfrak{p}_1\subsetneq \ldots \subsetneq \mathfrak{p}_n</math>.
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| For example, a field is zero-dimensional, since the only prime ideal is the zero ideal. It is also known that a commutative ring is Artinian if and only if it is Noetherian and zero-dimensional (i.e., all its prime ideals are maximal). The integers are one-dimensional: any chain of prime ideals is of the form
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| :<math>0 = \mathfrak p_0 \subsetneq p\mathbb Z = \mathfrak p_1</math>, where ''p'' is a [[prime number]]
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| since any ideal in '''Z''' is principal.
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| The dimension behaves well if the rings in question are Noetherian: the expected equality
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| :dim ''R''[''X''] = dim ''R'' + 1
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| holds in this case (in general, one has only dim ''R'' + 1 ≤ dim ''R''[''X''] ≤ 2 · dim ''R'' + 1). Furthermore, since the dimension depends only on one maximal chain, the dimension of ''R'' is the [[supremum]] of all dimensions of its localisations ''R''<sub>''p''</sub>, where ''p'' is an arbitrary prime ideal. Intuitively, the dimension of ''R'' is a local property of the spectrum of ''R''. Therefore, the dimension is often considered for local rings only, also since general Noetherian rings may still be infinite, despite all their localisations being finite-dimensional.
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| Determining the dimension of, say,
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| :''k''[''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub>] / (''f''<sub>1</sub>, ''f''<sub>2</sub>, ..., ''f''<sub>''m''</sub>), where ''k'' is a field and the ''f''<sub>''i''</sub> are some polynomials in ''n'' variables,
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| is generally not easy. For ''R'' Noetherian, the dimension of ''R'' / ''I'' is, by [[Krull's principal ideal theorem]], at least dim ''R'' − ''n'', if ''I'' is generated by ''n'' elements. If the dimension does drops as much as possible, i.e. dim ''R'' / ''I'' = dim ''R'' − ''n'', the ''R'' / ''I'' is called a [[complete intersection]].
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| A local ring ''R'', i.e. one with only one maximal ideal ''m'', is called [[regular local ring|regular]], if the (Krull) dimension of ''R'' equals the dimension (as a vector space over the field ''R'' / ''m'') of the cotangent space ''m'' / ''m''<sup>2</sup>.
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| <!--Another possible chain (which is more geometric) is the following chain of inclusions:
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| ''[[Cohen-Macaulay ring]]s'' ⊃ ''[[Gorenstein ring]]s'' ⊃ ''[[Regular ring]]s'' ⊃ ''[[Regular local ring]]s''
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| -->
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| == Constructing commutative rings ==
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| There are several ways to construct new rings out of given ones. The aim of such constructions is often to improve certain properties of the ring so as to make it more readily understandable. For example, an integral domain that is [[integrally closed]] in its [[field of fractions]] is called [[normal ring|normal]]. This is a desirable property, for example any normal one-dimensional ring is necessarily [[Regular local ring|regular]]. Rendering{{clarification needed|date=March 2012}} a ring normal is known as ''normalization''.
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| === Completions===
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| If ''I'' is an ideal in a commutative ring ''R'', the powers of ''I'' form [[neighborhood (topology)|topological neighborhoods]] of ''0'' which allow ''R'' to be viewed as a [[topological ring]]. This topology is called the [[I-adic topology|''I''-adic topology]]. ''R'' can then be completed with respect to this topology. Formally, the ''I''-adic completion is the [[inverse limit]] of the rings ''R''/''I<sup>n</sup>''. For example, if ''k'' is a field, ''k''<nowiki>[[</nowiki>''X''<nowiki>]]</nowiki>, the [[formal power series]] ring in one variable over ''k'', is the ''I''-adic completion of ''k''[''X''] where ''I'' is the principal ideal generated by ''X''. Analogously, the ring of ''p''-adic integers is the ''I''-adic completion of '''Z''' where ''I'' is the principal ideal generated by ''p''. Any ring that is isomorphic to its own completion, is called [[complete ring|complete]].
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| ==Properties==
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| By [[Wedderburn's little theorem|Wedderburn's theorem]], every finite [[division ring]] is commutative, and therefore a [[finite field]]. Another condition ensuring commutativity of a ring, due to [[Nathan Jacobson|Jacobson]], is the following: for every element ''r'' of ''R'' there exists an integer {{nowrap|''n'' > 1}} such that {{nowrap|1=''r''<sup>''n''</sup> = ''r''}}.<ref>{{Harvard citations|last = Jacobson|year = 1945|nb = yes}}</ref> If, ''r''<sup>2</sup> = ''r'' for every ''r'', the ring is called [[Boolean ring]]. More general conditions which guarantee commutativity of a ring are also known.<ref>{{Harvard citations|last = Pinter-Lucke|year = 2007|nb = yes}}</ref>
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| ==See also==
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| * [[Graded ring]]
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| * [[Almost commutative ring]]
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| * [[Almost ring]], a certain generalization of a commutative ring.
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| * [[Simplicial commutative ring]], a [[simplicial object]] in the category of commutative rings.
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| ==Notes==
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| <references group=nb/>
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| ===Citations===
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| <references />
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| ==References==
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| * {{Citation | last1=Atiyah | first1=Michael | author1-link=Michael Atiyah | last2=Macdonald | first2=I. G. | author2-link=Ian G. Macdonald | title=Introduction to commutative algebra | publisher=Addison-Wesley Publishing Co. | year=1969 }}
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| * {{Citation | last1=Balcerzyk | first1=Stanisław | last2=Józefiak | first2=Tadeusz | title=Commutative Noetherian and Krull rings | publisher=Ellis Horwood Ltd. | location=Chichester | series=Ellis Horwood Series: Mathematics and its Applications | isbn=978-0-13-155615-7 | year=1989}}
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| * {{Citation | last1=Balcerzyk | first1=Stanisław | last2=Józefiak | first2=Tadeusz | title=Dimension, multiplicity and homological methods | publisher=Ellis Horwood Ltd. | location=Chichester | series=Ellis Horwood Series: Mathematics and its Applications. | isbn=978-0-13-155623-2 | year=1989}}
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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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| * {{Citation | doi=10.2307/1969205 | last1=Jacobson | first1=Nathan | author1-link=Nathan Jacobson | title=Structure theory of algebraic algebras of bounded degree | year=1945 | journal=[[Annals of Mathematics]] | issn=0003-486X | volume=46 | issue=4 | pages=695–707 | jstor=1969205}}
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| * {{Citation | last1=Kaplansky | first1=Irving | author1-link=Irving Kaplansky | title=Commutative rings | publisher=[[University of Chicago Press]] | edition=Revised | mr=0345945 | year=1974}}
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| * {{Citation | last1=Matsumura | first1=Hideyuki | title=Commutative Ring Theory | publisher=[[Cambridge University Press]] | edition=2nd | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-36764-6 | year=1989}}
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| * {{Citation | last1=Nagata | first1=Masayoshi | author1-link=Masayoshi Nagata | title=Local rings | publisher=Interscience Publishers | series=Interscience Tracts in Pure and Applied Mathematics | isbn=978-0-88275-228-0 |year=1975 | mr=0155856 | origyear=1962 | volume=13 | pages=xiii+234}}
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| * {{Citation | last1=Pinter-Lucke | first1=James | title=Commutativity conditions for rings: 1950–2005 | doi=10.1016/j.exmath.2006.07.001 | year=2007 | journal=Expositiones Mathematicae | issn=0723-0869 | volume=25 | issue=2 | pages=165–174}}
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| * {{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | last2=Samuel | first2=Pierre | author2-link=Pierre Samuel | title=Commutative Algebra I, II | publisher= D. van Nostrand, Inc. | location=Princeton, N.J. | series=University series in Higher Mathematics | year=1958-60}} ''(Reprinted 1975-76 by Springer as volumes 28-29 of Graduate Texts in Mathematics.)''
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| [[Category:Commutative algebra]]
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| [[Category:Ring theory]]
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| [[Category:Algebraic structures]]
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