Editing Polynomial ring (section)

===Noncommutative polynomial rings===
{{Main|Free algebra}}

For polynomial rings of more than one variable, the products ''X''⋅''Y'' and ''Y''⋅''X'' are simply defined to be equal.  A more general notion of polynomial ring is obtained when the distinction between these two formal products is maintained.  Formally, the polynomial ring in ''n'' noncommuting variables with coefficients in the ring ''R'' is the [[monoid ring]] ''R''[''N''], where the monoid ''N'' is the [[free monoid]] on ''n'' letters, also known as the set of all strings over an alphabet of ''n'' symbols, with multiplication given by concatenation.  Neither the coefficients nor the variables need commute amongst themselves, but the coefficients and variables commute with each other.

Just as the polynomial ring in ''n'' variables with coefficients in the commutative ring ''R'' is the free commutative ''R''-algebra of rank ''n'', the noncommutative polynomial ring in ''n'' variables with coefficients in the commutative ring ''R'' is the free associative, unital ''R''-algebra on ''n'' generators, which is noncommutative when ''n''&nbsp;>&nbsp;1.