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| In [[abstract algebra]], '''extension of scalars''' is a means of producing a [[module (mathematics)|module]] over a [[ring (mathematics)|ring]] <math>S</math> from a module over another ring <math>R</math>, given a [[ring homomorphism|homomorphism]] <math>f : R \to S </math> between them. Intuitively, the new module admits multiplication by more scalars than the original one, hence the name ''extension''.
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| == Definition ==
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| In this definition the rings are assumed to be [[ring (mathematics)#Notes_on_the_definition|associative]], but not necessarily [[commutative ring|commutative]], or to have an [[identity element|identity]]. Also, modules are assumed to be [[left module]]s. The modifications needed in the case of right modules are straightforward.
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| Let <math>f : R \to S</math> be a homomorphism between two rings, and let <math>M</math> be a module over <math>R</math>. Consider the [[tensor product of modules|tensor product]] <math>_SM = S \otimes_R M</math>, where <math>S</math> is regarded as a right <math>R</math>-module via <math>f</math>. Since <math>S</math> is also a left module over itself, and the two actions commute, that is <math>s \cdot (s' \cdot r) = (s \cdot s') \cdot r</math> for <math>s,s' \in S</math>, <math>r \in R</math> (in a more formal language, <math>S</math> is a <math>(S,R)</math>-[[bimodule]]), <math>_SM</math> inherits a left action of <math>S</math>. It is given by <math>s \cdot (s' \otimes m) = ss' \otimes m</math> for <math>s,s' \in S</math> and <math>m \in M</math>. This module is said to be obtained from <math>M</math> through ''extension of scalars''.
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| Informally, extension of scalars is "the tensor product of a ring and a module"; more formally, it is a special case of a tensor product of a bimodule and a module – the tensor product of an <math>(S,R)</math> bimodule with an ''R''-module is an ''S''-module.
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| == Examples ==
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| One of the simplest examples is [[complexification]], which is extension of scalars from the [[real number]]s to the [[complex number]]s. More generally, given any [[field extension]] ''K'' < ''L,'' one can extend scalars from ''K'' to ''L.'' In the language of fields, a module over a field is called a [[vector space]], and thus extension of scalars converts a vector space over ''K'' to a vector space over ''L.'' This can also be done for [[division algebra]]s, as is done in [[quaternionification]] (extension from the reals to the [[quaternion]]s).
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| More generally, given a homomorphism from a field or ''commutative'' ring ''R'' to a ring ''S,'' the ring ''S'' can be thought of as an [[associative algebra]] over ''R,'' and thus when one extends scalars on an ''R''-module, the resulting module can be thought of alternatively as an ''S''-module, or as an ''R''-module with an [[algebra representation]] of ''S'' (as an ''R''-algebra). For example, the result of complexifying a real vector space (''R'' = '''R''', ''S'' = '''C''') can be interpreted either as a complex vector space (''S''-module) or as a real vector space with a [[linear complex structure]] (algebra representation of ''S'' as an ''R''-module).
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| === Applications ===
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| This generalization is useful even for the study of fields – notably, many algebraic objects associated to a field are not themselves fields, but are instead rings, such as algebras over a field, as in [[representation theory]]. Just as one can extend scalars on vector spaces, one can also extend scalars on [[group algebra]]s and also on modules over group algebras, i.e., [[group representation]]s. Particularly useful is relating how [[irreducible representation]]s change under extension of scalars – for example, the representation of the cyclic group of order 4, given by rotation of the plane by 90°, is an irreducible 2-dimensional ''real'' representation, but on extension of scalars to the complex numbers, it split into 2 complex representations of dimension 1. This corresponds to the fact that the [[characteristic polynomial]] of this operator, <math>x^2+1,</math> is irreducible of degree 2 over the reals, but factors into 2 factors of degree 1 over the complex numbers – it has no real eigenvalues, but 2 complex eigenvalues.
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| == Interpretation as a functor ==
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| Extension of scalars can be interpreted as a functor from <math>R</math>-modules to <math>S</math>-modules. It sends <math>M</math> to <math>_SM</math>, as above, and an <math>R</math>-homomorphism <math>u : M \to N</math> to the <math>S</math>-homomorphism <math>u_S : _SM \to _SN</math> defined by <math>u_S = \text{id}_S \otimes u</math>.
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| == Connection with restriction of scalars ==
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| Consider an <math>R</math>-module <math>M</math> and an <math>S</math>-module <math>N</math>. Given a homomorphism <math>u \in \text{Hom}_R(M,N)</math>, where <math>N</math> is viewed as an <math>R</math>-module via [[restriction of scalars]], define <math>Fu : _SM \to N</math> to be the [[function composition|composition]]
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| :<math>_SM = S \otimes_R M \xrightarrow{\text{id}_S \otimes u} S \otimes_R N \to N</math>,
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| where the last map is <math>s \otimes n \mapsto sn</math>. This <math>Fu</math> is an <math>S</math>-homomorphism, and hence <math>F : \text{Hom}_R(M,N) \to \text{Hom}_S(_SM,N)</math> is well-defined, and is a homomorphism (of [[abelian group]]s).
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| In case both <math>R</math> and <math>S</math> have an identity, there is an inverse homomorphism <math>G : \text{Hom}_S(_SM,N) \to \text{Hom}_R(M,N)</math>, which is defined as follows. Let <math>v \in \text{Hom}_S(_SM,N)</math>. Then <math>Gv</math> is the composition
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| :<math>M \to R \otimes_R M \xrightarrow{f \otimes \text{id}_M} S \otimes_R M \xrightarrow{v} N</math>,
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| where the first map is the [[canonical form|canonical]] [[isomorphism]] <math>m \mapsto 1 \otimes m</math>.
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| This construction shows that the groups <math>\text{Hom}_S(_SM,N)</math> and <math>\text{Hom}_R(M,N)</math> are isomorphic. Actually, this isomorphism depends only on the homomorphism <math>f</math>, and so is [[functorial]]. In the language of [[category theory]], the extension of scalars functor is [[left adjoint]] to the restriction of scalars functor.
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| == See also ==
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| * [[Restriction of scalars]]
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| * [[Tensor product of fields]]
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| == References ==
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| NICOLAS BOURBAKI. Algebra I, Chapter II. LINEAR ALGEBRA.§5. Extension of the ring of scalaxs;§7. Vector spaces. 1974 by Hermann.
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| [[Category:Module theory]]
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