A left hereditary ring is a ring in which every left ideal is a projective module. Equivalently, every submodule of a projective left module is projective. Right hereditary rings are defined with right ideals and modules instead (Wisbauer 1991, pp. 328, 338-339). The left and right conditions need not coincide. Upper triangular matrix rings over a field satisfy both conditions (Wisbauer 1991, p. 339).
Hereditary Ring
See also
Left Ideal, Projective Module, Right Ideal, Ring, SubmoduleExplore with Wolfram|Alpha
References
Wisbauer, R. Foundations of Module and Ring Theory: A Handbook for Study and Research. Reading: Gordon and Breach, pp. 328, 338-339, 1991. https://www.math.uni-duesseldorf.de/~wisbauer/book.pdf.Cite this as:
Weisstein, Eric W. "Hereditary Ring." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HereditaryRing.html