Triangular matrix ring

In algebra, a triangular matrix ring, also called a triangular ring, is a ring constructed from two rings and a bimodule.
Definition.
If formula_1 and formula_2 are rings and formula_3 is a formula_4-bimodule, then the triangular matrix ring formula_5 consists of 2 by 2 matrices of the form formula_6, where formula_7 and formula_8 with ordinary matrix addition and matrix multiplication as its operations.