Editing Group ring (section)

=== Universal property ===
The above adjunction expresses a universal property of group rings.<ref name=Milies/><ref>{{Cite web|url=https://ncatlab.org/nlab/show/group+algebra#general|title=group algebra in nLab|website=ncatlab.org|access-date=2017-11-01}}</ref> Let {{var|R}} be a  (commutative) ring, let {{var|G}} be a group, and let {{var|S}} be an {{var|R}}-algebra. For any group homomorphism <math>f:G\to S^\times</math>, there exists a unique {{var|R}}-algebra homomorphism <math>\overline{f}:R[G]\to S</math> such that <math>\overline{f}^\times \circ i=f</math> where {{var|i}} is the inclusion

:<math>\begin{align}
  i:G &\longrightarrow R[G] \\
  g &\longmapsto 1_Rg
\end{align}</math>

In other words, <math>\overline{f}</math> is the unique homomorphism making the following diagram commute:

:[[Image:Group ring UMP.svg|200px]]

Any other ring satisfying this property is [[List of mathematical jargon|canonically]] isomorphic to the group ring.