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is balanced, and the subgroup has been chosen minimally so that this map is balanced. The universal property of ⊗ follows from the universal properties of a free abelian group and a quotient.
If ''S'' is a subring of a ring ''R'', then <math>M \otimes_R N</math> is the quotient group of <math>M \otimes_S N</math> by the subgroup generated by <math>xr \otimes_S y - x \otimes_S ry, \, r \in R, x \in M, y \in N</math>, where <math>x \otimes_S y</math> is the image of <math>(x, y)</math> under <math>\otimes: M \times N \to M \otimes_{S} N.</math> In particular, any tensor product of ''R''-modules can be constructed, if so desired, as a quotient of a tensor product of abelian groups by imposing the ''R''-balanced product property.▼
More category-theoretically, let σ be the given right action of ''R'' on ''M''; i.e., σ(''m'', ''r'') = ''m'' · ''r'' and τ the left action of ''R'' of ''N''. Then, provided the tensor product of abelian groups is already defined, the tensor product of ''M'' and ''N'' over ''R'' can be defined as the [[coequalizer]]:
<math display="block">M \otimes R \otimes N {{{} \atop \overset{\sigma \times 1}\to}\atop{\underset{1 \times \tau} \to \atop {}}} M \otimes N \overset{\otimes}\to M \otimes_R N,</math>
▲If ''S'' is a subring of a ring ''R'', then <math>M \otimes_R N</math> is the quotient group of <math>M \otimes_S N</math> by the subgroup generated by <math>xr \otimes_S y - x \otimes_S ry, \, r \in R, x \in M, y \in N</math>, where <math>x \otimes_S y</math> is the image of <math>(x, y)</math> under <math>\otimes: M \times N \to M \otimes_{S} N.</math> In particular, any tensor product of ''R''-modules can be constructed, if so desired, as a quotient of a tensor product of abelian groups by imposing the ''R''-balanced product property.
In the construction of the tensor product over a commutative ring ''R'', the ''R''-module structure can be built in from the start by forming the quotient of a free ''R''-module by the submodule generated by the elements given above for the general construction, augmented by the elements {{math|''r'' ⋅ (''m'' ∗ ''n'') − ''m'' ∗ (''r'' ⋅ ''n'')}}. Alternately, the general construction can be given a Z(''R'')-module structure by defining the scalar action by {{math|1=''r'' ⋅ (''m'' ⊗ ''n'') = ''m'' ⊗ (''r'' ⋅ ''n'')}} when this is well-defined, which is precisely when ''r'' ∈ Z(''R''), the [[Center (ring theory)|centre]] of ''R''.
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