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 in particular left semihereditary. Von Neumann regular rings are semihereditary, since their finitely generated ideals are generated by idempotents and are projective. In the commutative setting, semihereditary means every finitely generated ideal is projective. This includes principal ideal domains and Dedekind domains, which are in fact hereditary in the commutative sense. Semisimple rings also satisfy the condition because every module is projective.

Coherence and structural implications: A left (or right) semihereditary ring is left (or right) coherent, because finitely generated projective modules are finitely presented, making every finitely generated ideal finitely presented.

Examples and scope: Principal ideal domains and Dedekind domains are classic examples in the commutative case. In general, semihereditary rings form a useful class in noncommutative ring theory, connecting ideal-theoretic properties with module-theoretic behavior. The concept highlights rings in which projectivity of finitely generated ideals yields tighter control over the structure of modules and extensions.