# Semi-hereditary ring

A ring each finitely-generated left ideal of which is projective (cf. also Projective module). Examples are the ring of integers, the ring of polynomials in one variable over a field, von Neumann regular rings (cf. Regular ring (in the sense of von Neumann)), hereditary rings, rings of finitely-generated free ideals (semi-FI-ring). An analogous definition yields right semi-hereditary rings. A left semi-hereditary ring is not necessarily right semi-hereditary. However, a local left semi-hereditary ring is an integral domain and a right semi-hereditary ring. A ring of matrices over a semi-hereditary ring is semi-hereditary. If $R$ is a semi-hereditary ring and there is an $e\in R$ with $e^2=e$, then $eRe$ is a semi-hereditary ring. A finitely-generated submodule of a projective module over a semi-hereditary ring is isomorphic to the direct sum of a certain set of finitely-generated left ideals of the ground ring; consequently, it is projective. Each such module can also be represented as a direct sum of modules dual to finitely-generated right ideals of the ground ring.
For a commutative ring $R$, the following properties are equivalent: 1) $R$ is semi-hereditary; 2) $(A\cap B)C=AC\cap BC$, where $A$, $B$ and $C$ are arbitrary ideals in $R$; 3) the complete ring of fractions of $R$ is regular in the sense of von Neumann, and for every maximal ideal $\mathfrak m$ of $R$ the ring of fractions $R_\mathfrak m$ is a normal ring; and 4) all $2$-generated ideals of $R$ are projective. The ring of polynomials in one variable over a commutative ring $T$ is semi-hereditary if and only if $R$ is regular in the sense of von Neumann.