Editing Modular representation theory (section)

==Reduction (mod ''p'')==
In the theory initially developed by Brauer, the link between ordinary representation theory and modular representation theory is best exemplified by considering the
[[group ring|group algebra]] of the group ''G'' over a complete [[discrete valuation ring]] ''R'' with [[residue field]] ''K'' of positive
characteristic ''p'' and field of fractions ''F'' of characteristic
0, such as the [[p-adic number#p-adic integers|''p''-adic integers]]. The structure of ''R''[''G''] is closely related both to
the structure of the group algebra ''K''[''G''] and to the structure of the semisimple group algebra  ''F''[''G''], and there is much interplay
between the module theory of the three algebras.

Each ''R''[''G'']-module naturally gives rise to an ''F''[''G'']-module, and, by a process often known informally as '''reduction (mod ''p'')''',
to a ''K''[''G'']-module. On the other hand, since ''R'' is a [[principal ideal domain]], each finite-dimensional ''F''[''G'']-module
arises by extension of scalars from an ''R''[''G'']-module.{{citation needed|date=August 2024}} In general, however, not all ''K''[''G'']-modules arise as reductions (mod ''p'') of
''R''[''G'']-modules. Those that do are '''liftable'''.