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| {{about|the algebraic group ring of a discrete group|the case of a topological group|group algebra|a general group|Group Hopf algebra}}
| | You could have seen an adhesive tool that is in clear plastic form. This is actually called epoxy resin. It has two parts; the liquid hardener and the powder. Those two should be blended properly so the clear plastic will be created. When this is mixed, it must be employed right away since it has the tendency to harden quickly. This can be used on wood, metal, plastic and many many other materials. When working with epoxy resin, it is important that one knows how to properly put it to use in order that it will not create difficulties and headaches to its users. Follow this advice on how to make use of this the right way.<br><br>For those who have any kind of concerns concerning where by as well as the best way to utilize [http://wilfordb1.deviantart.com/journal/Prestaciones-De-Los-Elementos-Reforzados-Con-Fibra-460898782 resina epoxi], you possibly can email us from our web page. Keep in mind that you need to use the epoxy resin onto floors that are neat and dry. Wet surface won't make certain robust bond between the epoxy as well as material. It's also wise to use this on a typical temperature. An excessive amount of heat or an excessive amount of cold will affect the efficiency of the epoxy. Before, you mix the components of the epoxy resin make sure that the top you will be gluing is already well prepared. It's also advisable to stick to instructions cautiously. Determine your liquid hardener correctly so your mixture will not be a failure.<br><br>It is best to have an estimate of just how much you will be utilizing for the task. That will even be the quantity that you need to mix. Excessive combination will not be useful in the long term. An estimation of just how much you will end up using will allow you to conserve much on your resin and steer clear of purchasing a different one very often. When epoxy is included large amounts, it can give off fumes which can be combustible and dangerous. And excess mixture will tend to harden quickly that will not be appropriate any more.<br><br>You might want to utilize wooden spoon to blend the liquid and powder together. Make sure that it is well mixed and what you are mixing will be sufficient for the project that you will be carrying out. Quit mixing when you feel that the mixture is beginning to get hardened. Make use of the mixture right away as it will firm up within 15 minutes. In case the temperatures are hot, the mixture are going to harden. This is why after the mixture is completed, this should actually be utilized immediately.<br><br>Epoxy resin is a very useful and flexible adhesive. It can be used on numerous projects. This results in a very good bond when applied to metal or wood. This is the reason most carpenters and also wood workers prefer by using this on their jobs. You can be sure that the bond this tool produces is tough and can be very durable. This doesn't very easily break and won't be affected by any temperature as soon as it is utilized.<br><br>You can buy this tool in hardware stores around. But you may also turn to websites which can be offering adhesive products. Hold It is one website that carries all kinds of adhesives that individuals make need. They offer online acquiring to people who are too busy to go out and buy the tool. People can easily purchase just by browsing this website. People can buy the various adhesive products they want and store it in their places for future use. |
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| In [[algebra]], a '''group ring''' is a [[free module]] and at the same time a [[Ring (mathematics)|ring]], constructed in a natural way from any given ring and any given [[Group (mathematics)|group]]. As a free module, its ring of scalars is the given ring, and its basis is one-to-one with the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring.
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| If the given ring is commutative, a group ring is also referred to as a [[group algebra]], for it is indeed an [[Algebra (ring theory)|algebra]] over the given ring.
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| The apparatus of group rings is especially useful in the theory of [[group representation]]s.
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| ==Definition==
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| Let ''G'' be a group, written multiplicatively, and let ''R'' be a ring. The group ring of ''G'' over ''R'', which we will denote by ''R''[''G''], is the set of mappings ''f'' : ''G'' → ''R'' of [[Support (mathematics)|finite support]],<ref name="Polcino">Polcino & Sehgal (2002), p. 131.</ref> where the product α''f'' of a scalar α in ''R'' and a vector (or mapping) ''f'' is defined as the vector <math>x \mapsto \alpha \cdot f(x)</math>, and the sum of two vectors ''f'' and ''g'' is defined as the vector <math>x \mapsto f(x) + g(x)</math>. To turn the additive group ''R''[''G''] into a ring, we define the product of ''f'' and ''g'' to be the vector
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| :<math>x\mapsto\sum_{uv=x}f(u)g(v)=\sum_{u\in G}f(u)g(u^{-1}x).</math>
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| The summation is legitimate because ''f'' and ''g'' are of finite support, and the ring axioms are readily verified.
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| Some variations in the notation and terminology are in use. In particular, the mappings such as ''f'' : ''G'' → ''R'' are sometimes written as what are called "formal linear combinations of elements of ''G'', with coefficients in ''R''":<ref>Polcino & Sehgal (2002), p. 129 and 131.</ref>
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| :<math>\sum_{g\in G}f(g) g,</math>
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| or simply
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| :<math>\sum_{g\in G}f_g g,</math>
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| where this doesn't cause confusion.<ref name="Polcino" />
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| ==Two simple examples==
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| Let ''G'' = '''Z'''<sub>3</sub>, the [[cyclic group]] of three elements with generator ''a'' and identity element 1<sub>''G''</sub>. An element ''r'' of '''C'''[''G''] may be written as
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| :<math>r = z_0 1_G + z_1 a + z_2 a^2\,</math>
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| where ''z''<sub>0</sub>, ''z''<sub>1</sub> and ''z''<sub>2</sub> are in '''C''', the [[complex numbers]]. Writing a different element ''s'' as
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| :<math>s=w_0 1_G +w_1 a +w_2 a^2\,</math> | |
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| their sum is
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| :<math>r + s = (z_0+w_0) 1_G + (z_1+w_1) a + (z_2+w_2) a^2\,</math>
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| and their product is
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| :<math>rs = (z_0w_0 + z_1w_2 + z_2w_1) 1_G +(z_0w_1 + z_1w_0 + z_2w_2)a +(z_0w_2 + z_2w_0 + z_1w_1)a^2.</math>
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| Notice that the identity element 1<sub>''G''</sub> of ''G'' induces a canonical embedding of the coefficient ring (in this case '''C''') into '''C'''[''G'']; however strictly speaking the multiplicative identity element of '''C'''[''G''] is 1⋅1<sub>''G''</sub> where the first ''1'' comes from '''C''' and the second from ''G''. The additive identity element is of course zero.
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| When ''G'' is a non-commutative group, one must be careful to preserve the order of the group elements (and not accidentally commute them) when multiplying the terms.
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| A different example is that of the [[Laurent polynomial]]s over a ring ''R'': these are nothing more or less than the group ring of the [[infinite cyclic group]] '''Z''' over ''R''.
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| ==Some basic properties==
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| Assuming that the ring ''R'' has a unit element 1, and denoting the group unit by 1<sub>''G''</sub>, the ring ''R''[''G''] contains a subring isomorphic to ''R'', and its group of invertible elements contains a subgroup isomorphic to ''G''. For considering the [[indicator function]] of {1<sub>''G''</sub>}, which is the vector ''f'' defined by
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| :<math>f(g)= 1\cdot 1_G + \sum_{g\not= 1_G}0 \cdot g= \mathbf{1}_{\{1_G\}}(g)=\begin{cases}
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| 1 & g = 1_G \\
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| 0 & g \ne 1_G
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| \end{cases},</math>
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| the set of all scalar multiples of ''f'' is a subring of ''R''[''G''] isomorphic to ''R''. And if we map each element ''s'' of ''G'' to the indicator function of {''s''}, which is the vector ''f'' defined by
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| :<math>f(g)= 1\cdot s + \sum_{g\not= s}0 \cdot g= \mathbf{1}_{\{s\}}(g)=\begin{cases}
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| 1 & g = s \\
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| 0 & g \ne s
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| \end{cases}</math>
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| the resulting mapping is an injective group homomorphism (with respect to multiplication, not addition, in ''R''[''G'']).
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| If ''R'' and ''G'' are both commutative (i.e., ''R'' is commutative and ''G'' is an [[abelian group]]), ''R''[''G''] is commutative.
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| If ''H'' is a [[subgroup]] of ''G'', then ''R''[''H''] is a [[subring]] of ''R''[''G'']. Similarly, if ''S'' is a subring of ''R'', ''S''[''G''] is a subring of ''R''[''G''].
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| ==Group algebra over a finite group==
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| Group algebras occur naturally in the theory of [[group representation]]s of [[finite group]]s. The group algebra ''K''[''G''] over a field ''K'' is essentially the group ring, with the field ''K'' taking the place of the ring. As a set and vector space, it is the [[free vector space]] on ''G'' over the field ''K''. That is, for ''x'' in ''K''[''G''],
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| :<math>x=\sum_{g\in G} a_g g.</math>
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| The [[algebra over a field|algebra]] structure on the vector space is defined using the multiplication in the group:
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| :<math>g \cdot h = gh,</math>
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| where on the left, ''g'' and ''h'' indicate elements of the group algebra, while the multiplication on the right is the group operation (denoted by juxtaposition).
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| Because the above multiplication can be confusing, one can also write the [[basis vector]]s of ''K''[''G''] as ''e''<sub>''g''</sub> (instead of ''g''), in which case the multiplication is written as:
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| :<math>e_g \cdot e_h = e_{gh}.</math>
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| ===Interpretation as functions===
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| Thinking of the [[free vector space]] as ''K''-valued functions on ''G'', the algebra multiplication is [[convolution]] of functions.
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| While the group algebra of a ''finite'' group can be identified with the space of functions on the group, for an infinite group these are different. The group algebra, consisting of ''finite'' sums, corresponds to functions on the group that vanish for [[cofinitely]] many points; topologically (using the [[discrete topology]]), these correspond to functions with [[compact support]].
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| However, the group algebra ''K''[''G''] and the space of functions ''K''<sup>''G''</sup> := Hom(''G'',''K'') are dual: given an element of the group algebra
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| :<math>x = \sum_{g\in G} a_g g</math>
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| and a function on the group ''f'' : ''G'' → ''K'' these pair to give an element of ''K'' via
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| :<math>(x,f) = \sum_{g\in G} a_g f(g),</math>
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| which is a well-defined sum because it is finite.
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| ===Regular representation===
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| The group algebra is an algebra over itself; under the correspondence of representations over ''R'' and ''R''[''G''] modules, it is the [[regular representation]] of the group.
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| Written as a representation, it is the representation ''g'' {{mapsto}} ρ<sub>''g''</sub> with the action given by <math>\rho(g)\cdot e_h = e_{gh}</math>, or
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| :<math>\rho(g)\cdot r = \sum_{h\in G} k_h \rho(g)\cdot e_h = \sum_{h\in G} k_h e_{gh}. </math>
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| ===Properties===
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| The dimension of the vector space ''K''[''G''] is just equal to the number of elements in the group. The field ''K'' is commonly taken to be the complex numbers '''C''' or the reals '''R''', so that one discusses the group algebras '''C'''[''G''] or '''R'''[''G''].
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| The group algebra '''C'''[''G''] of a finite group over the complex numbers is a [[semisimple ring]]. This result, [[Maschke's theorem]], allows us to understand '''C'''[''G''] as a finite [[Product of rings|product]] of [[matrix ring]]s with entries in '''C'''.
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| Group rings satisfy a [[universal property]].<ref name="Polcino" />
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| ===Representations of a group algebra===
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| Taking ''K''[''G''] to be an abstract algebra, one may ask for concrete [[group representation|representations]] of the algebra over a vector space ''V''. Such a representation
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| :<math>\tilde{\rho}:K[G]\rightarrow \mbox{End} (V).</math>
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| is an algebra homomorphism from the group algebra to the set of [[endomorphism]]s on ''V''. Taking ''V'' to be an [[abelian group]], with group addition given by vector addition, such a representation is in fact a [[module (mathematics)|left ''K''[''G'']-module]] over the abelian group ''V''. This is demonstrated below, where each axiom of a module is confirmed.
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| Pick ''r'' ∈ ''K''[''G''] so that
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| :<math>\tilde{\rho}(r) \in \mbox{End}(V).</math>
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| Then <math>\tilde{\rho}(r)</math> is a homomorphism of abelian groups, in that
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| :<math>\tilde{\rho}(r) \cdot (v_1 +v_2) = \tilde{\rho}(r) \cdot v_1 + \tilde{\rho}(r) \cdot v_2</math>
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| for any ''v''<sub>1</sub>, ''v''<sub>2</sub> ∈ ''V''. Next, one notes that the set of endomorphisms of an abelian group is an [[endomorphism ring]]. The representation <math>\tilde{\rho}</math> is a ring homomorphism, in that one has
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| :<math>\tilde{\rho}(r+s)\cdot v = \tilde{\rho}(r)\cdot v + \tilde{\rho}(s)\cdot v </math>
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| for any two ''r'', ''s'' ∈ ''K''[''G''] and ''v'' ∈ ''V''. Similarly, under multiplication,
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| :<math>\tilde{\rho}(rs)\cdot v = \tilde{\rho}(r)\cdot \tilde{\rho}(s)\cdot v. </math>
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| Finally, one has that the unit is mapped to the identity:
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| :<math>\tilde{\rho}(1)\cdot v = v</math>
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| where 1 is the multiplicative unit of ''K''[''G'']; that is,
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| : <math>1 = e_e</math>
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| is the vector corresponding to the identity element ''e'' in ''G''.
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| The last three equations show that <math>\tilde{\rho}</math> is a ring homomorphism from ''K''[''G''] taken as a group ring, to the endomorphism ring. The first identity showed that individual elements are group homomorphisms. Thus, a representation <math>\tilde{\rho}</math> is a left ''K''[''G'']-module over the abelian group ''V''.
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| Note that given a general ''K''[''G'']-module, a vector-space structure is induced on ''V'', in that one has an additional axiom
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| :<math> \tilde{\rho}(ar) \cdot v_1 + \tilde{\rho}(br) \cdot v_2 = a \tilde{\rho}(r) \cdot v_1 + b \tilde{\rho}(r) \cdot v_2 = \tilde{\rho}(r) \cdot (av_1 +bv_2)</math>
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| for scalar ''a'', ''b'' ∈ ''K''.
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| Any group representation
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| :<math>\rho:G\rightarrow \mbox{Aut}(V),</math>
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| with ''V'' a vector space over the field ''K'', can be extended linearly to an algebra representation
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| :<math>\tilde{\rho}:K[G]\rightarrow \mbox{End}(V),</math>
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| simply by mapping <math>\rho(g) \mapsto \tilde{\rho}(e_g)</math>. Thus, representations of the group correspond exactly to representations of the algebra, and so, in a certain sense, talking about the one is the same as talking about the other.
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| ===Center of a group algebra===
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| The [[center of a group|center]] of the group algebra is the set of elements that commute with all elements of the group algebra:
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| :<math>Z(K[G]) := \left\{ z \in K[G] \ : \ \forall r \in K[G], zr = rz \right\}.</math>
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| The center is equal to the set of [[class function]]s, that is the set of elements that are constant on each conjugacy class
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| :<math>Z(K[G]) = \left\{ \sum_{g \in G} a_g g \ : \ \forall g,h \in G, a_g = a_{h^{-1}gh}\right\}.</math>
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| If {{nowrap|1=''K'' = '''C'''}}, the set of irreducible [[character theory|characters]] of ''G'' forms an orthonormal basis of ''Z''(''K''[''G'']) with respect to the inner product
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| :<math>\left \langle \sum_{g \in G} a_g g, \sum_{g \in G} b_g g \right \rangle = \frac{1}{|G|} \sum_{g \in G} \bar{a}_g b_g.</math>
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| ==Group rings over an infinite group==
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| Much less is known in the case where ''G'' is countably infinite, or uncountable, and this is an area of active research. The case where ''R'' is the field of complex numbers is probably the one best studied. In this case, [[Irving Kaplansky]] proved that if ''a'' and ''b'' are elements of '''C'''[''G''] with ''ab'' = 1, then ''ba'' = 1. Whether this is true if ''R'' is a field of positive characteristic remains unknown.
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| A long-standing conjecture of Kaplansky (~1940) says that if ''G'' is a [[torsion-free group]], and ''K'' is a field, then the group ring ''K''[''G''] has no non-trivial [[zero divisor]]s. This conjecture is equivalent to ''K''[''G''] having no non-trivial [[nilpotent]]s under the same hypotheses for ''K'' and ''G''.
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| In fact, the condition that ''K'' is a field can be relaxed to any ring that can be embedded into an [[integral domain]].
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| The conjecture remains open in full generality, however some special cases of torsion-free groups have been shown to satisfy the zero divisor conjecture. These include:
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| * Unique product groups (which include [[virtually abelian group]]s, [[orderable group]]s, and [[free group]]s, since they are orderable)
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| * [[Elementary amenable group]]s
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| * Diffuse groups - in particular, groups that act freely isometrically on ''R''-trees, and the fundamental groups of surface groups except for the fundamental groups of direct sums of one, two or three copies of the projective plane.
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| The case of ''G'' being a [[topological group]] is discussed in greater detail in the article on [[group algebra]]s.
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| ==Representations of a group ring==
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| A module ''M'' over ''R''[''G''] is then the same as a [[group representation|linear representation]] of ''G'' over the field ''R''. There is no particular reason to limit ''R'' to be a field here. However, the classical results were obtained first when ''R'' is the [[complex number]] field and ''G'' is a finite group, so this case deserves close attention. It was shown that ''R''[''G''] is a [[semisimple ring]], under those conditions, with profound implications for the representations of finite groups. More generally, whenever the [[characteristic (algebra)|characteristic]] of the field ''R'' does not divide the order of the finite group ''G'', then ''R''[''G''] is semisimple ([[Maschke's theorem]]).
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| When ''G'' is a finite [[abelian group]], the group ring is commutative, and its structure is easy to express in terms of [[root of unity|roots of unity]]. When ''R'' is a field of characteristic ''p'', and the prime number ''p'' divides the order of the finite group ''G'', then the group ring is ''not'' semisimple: it has a non-zero [[Jacobson radical]], and this gives the corresponding subject of [[modular representation theory]] its own, deeper character.
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| ==Category theory==
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| ===Adjoint===
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| [[Category theory|Categorically]], the group ring construction is [[left adjoint]] to "[[group of units]]"; the following functors are an [[adjoint functors|adjoint pair]]:
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| :<math>\operatorname{GrpRng}\colon \mathbf{\operatorname{Grp}} \to R\mathbf{\operatorname{-Alg}}</math>
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| :<math>\operatorname{GrpUnits}\colon R\mathbf{\operatorname{-Alg}} \to \mathbf{\operatorname{Grp}}</math>
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| where "GrpRng" takes a group to its group ring over ''R'', and "GrpUnits" takes an ''R''-algebra to its group of units.
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| When ''R'' = '''Z''', this gives an adjunction between the [[category of groups]] and the [[category of rings]], and the unit of the adjunction takes a group ''G'' to a group that contains trivial units: ''G'' × {±1} = {±''g''}. In general, group rings contain nontrivial units. If ''G'' contains elements ''a'' and ''b'' such that <math>a^n=1</math> and ''b'' does not normalize <math>\langle a\rangle</math> then the square of
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| :<math>x=(a-1)b \left (1+a+a^2+...+a^{n-1} \right )</math>
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| is zero, hence <math>(1+x)(1-x)=1</math>. The element 1+''x'' is a unit of infinite order.
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| ===Generalizations===
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| The group algebra generalizes to the [[monoid ring]] and thence to the [[categorical algebra]], of which another example is the [[incidence algebra]].
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| ==Filtration==
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| {{Expand section|date=December 2008}}
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| If a group has a [[length function]] – for example, if there is a choice of generators and one takes the [[word metric]], as in [[Coxeter group]]s – then the group ring becomes a [[filtered algebra]].
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| ==See also==
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| * [[Group algebra]]
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| * [[Monoid ring]]
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| ===Representation theory===
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| * [[Group representation]]
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| * [[Regular representation]]
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| ===Category theory===
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| * [[Categorical algebra]]
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| * [[Group of units]]
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| * [[Incidence algebra]]
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| == Notes ==
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| {{Reflist}}
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| ==References==
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| * {{springer|id=G/g045220|title=Group algebra|author=A. A. Bovdi}}
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| * Milies, César Polcino; Sehgal, Sudarshan K.. ''An introduction to group rings''. Algebras and applications, Volume 1. Springer, 2002. ISBN 978-1-4020-0238-0
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| * [[Charles W. Curtis]], [[Irving Reiner]], ''Representation theory of finite groups and associative algebras'', Interscience (1962)
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| * D.S. Passman, ''The algebraic structure of group rings'', Wiley (1977)
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| {{DEFAULTSORT:Group Ring}}
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| [[Category:Ring theory]]
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| [[Category:Representation theory of groups]]
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| [[Category:Harmonic analysis]]
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| [[de:Monoidring]]
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