Hereditary ring

Definition
A commutative unital ring is termed hereditary if it satisfies the following equivalent conditions:


 * Every ideal in it is a projective module
 * Every submodule of a free module is projective
 * Every submodule of a projective module is projective
 * Every quotient of an injective module is injective
 * The global dimension of the ring is at most one

Stronger properties

 * Field
 * Dedekind domain: In fact, an integral domain is hereditary if and only if it is a Dedekind domain
 * Semisimple Artinian ring: This is a ring with global dimension zero

Weaker properties

 * Semihereditary ring