semihereditary
Semihereditary is a term used in ring theory to describe rings in which finitely generated ideals are projective. More precisely, a ring R is left semihereditary if every finitely generated left ideal of R is projective as a left R-module. The analogous right notion uses right ideals. A ring that is both left and right semihereditary is commonly referred to simply as semihereditary, with the understanding that the property may be specified as left or right when appropriate.
Key relationships and consequences: If a ring is left hereditary (every left ideal is projective), it is
Coherence and structural implications: A left (or right) semihereditary ring is left (or right) coherent, because
Examples and scope: Principal ideal domains and Dedekind domains are classic examples in the commutative case.