Suppose that and are two algebras and is a unital -bimodule. Then
with the usual matrix-like addition and matrix-like multiplication is an algebra.
An algebra is called a triangular algebra if there exist algebras and and an -bimodule such that is (algebraically) isomorphic to
under matrix-like addition and matrix-like multiplication.
For example, the algebra of upper triangular matrices over the complex field may be viewed as a triangular algebra when .