Noncommutative algebra is the study of algebras in which multiplication need not be commutative. Thus elements
and 
may satisfy 
,
and the order of factors must be retained in formulas. Algebras of matrices,
operator algebras, group
algebras, and tensor algebras are basic sources
of noncommutative algebras.
Many constructions from commutative algebra have left and right versions in the noncommutative setting. For example, left ideals and right ideals need not coincide, and the order of factors matters when forming polynomials or fractions.
