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==Additional structure==
{{confusing|The whole paragraph at the end is confusing. Also it seems to repeat what is already mentioned earlier.|date=July 2022}}
{{see also|Free product of associative algebras}}
If ''S'' and ''T'' are commutative ''R''-algebras, then, similar to [[#For equivalent modules]], {{math|''S'' ⊗<sub>''R''</sub> ''T''}} will be a commutative ''R''-algebra as well, with the multiplication map defined by {{math|1=(''m''<sub>1</sub> ⊗ ''m''<sub>2</sub>) (''n''<sub>1</sub> ⊗ ''n''<sub>2</sub>) = (''m''<sub>1</sub>''n''<sub>1</sub> ⊗ ''m''<sub>2</sub>''n''<sub>2</sub>)}} and extended by linearity. In this setting, the tensor product become a [[fibered coproduct]] in the category of commutative ''R''-algebras. (But it is not a coproduct in the category of ''R''-algebras.) <!--Note that any ring is a '''Z'''-algebra, so we may always take {{math|''M'' ⊗<sub>'''Z'''</sub> ''N''}}.-->
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