matrix ring is right hereditary and left semi-hereditary but not left hereditary. If S is a von Neumann regular ring with an ideal I that is not a direct summand, then the triangular matrix ring is left semi-hereditary but not right semi-hereditary. Properties For a left hereditary ring R, every submodule of a free left R-module is isomorphic to a direct sum of left ideals of R and hence is projective. References Ring theory