Hereditary ring

In mathematics, the length of a projective resolution of a module over a ring provides in some sense a measure of how the module is " complicated".

A ring is called hereditary if every submodule of a projective module is projective. That is, any minimal projective resolution of a module stops after only two steps.

For non - commutative rings, a distinction between left and Rechtserblichkeit: A ring is called left - hereditary, if every submodule of a projective module is projective links. Accordingly, a ring is called right - hereditary, if every submodule of a projective module is projective - law. There are rings that are left - but not right - hereditary, and vice versa.

Examples

• Every body is hereditary, because all -modules ( = - vector spaces ) are free and thus projective.
• Every semisimple ring is hereditary, because each module over the ring is projective.
• Every principal ideal ring is hereditary, since projective modules are free and submodules of free modules also free.
• Each path algebra of a quiver is hereditary.

• Algebra