${\displaystyle {}R}$

${\displaystyle M\longrightarrow R'\otimes _{R}M,\,v\longmapsto 1\otimes v.}$

Bei ${\displaystyle {}R=R'}$ ist dies ein Isomorphismus.

${\displaystyle {}R}$

${\displaystyle {}\varphi \colon M\rightarrow N}$

${\displaystyle \operatorname {Id} _{R'}\otimes \varphi \colon R'\otimes _{R}M\longrightarrow R'\otimes _{R}N}$

ein ${\displaystyle {}R'}$-Modulhomomorphismus.

${\displaystyle {}M=R^{n}}$

${\displaystyle {}R'\otimes _{R}R^{n}\cong {\left(R'\right)}^{n}\,.}$

${\displaystyle {}R'\rightarrow R^{\prime \prime }}$

${\displaystyle {}R^{\prime \prime }\otimes _{R}M\cong R^{\prime \prime }\otimes _{R'}{\left(R^{\prime }\otimes _{R}M\right)}\,}$

(eine Isomorphie von ${\displaystyle R^{\prime \prime }}$-Moduln).