View source for Semi-hereditary ring

''from the left''

A [[Ring|ring]] each finitely-generated left ideal of which is projective (cf. also [[Projective module|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)|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s0841601.png" /> is a semi-hereditary ring and there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s0841602.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s0841603.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s0841604.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s0841605.png" />, the following properties are equivalent: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s0841606.png" /> is semi-hereditary; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s0841607.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s0841608.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s0841609.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s08416010.png" /> are arbitrary ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s08416011.png" />; 3) the complete ring of fractions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s08416012.png" /> is regular in the sense of von Neumann, and for every maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s08416013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s08416014.png" /> the ring of fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s08416015.png" /> is a normal ring; and 4) all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s08416016.png" />-generated ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s08416017.png" /> are projective. The ring of polynomials in one variable over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s08416018.png" /> is semi-hereditary if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084160/s08416019.png" /> is regular in the sense of von Neumann.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cartan,   S. Eilenberg,   "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  L.A. Skornyakov,   A.V. Mikhalev,   "Modules"  ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''14'''  (1976)  pp. 57–190  (In Russian)</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  V.T. Markov,   A.V. Mikhalev,   L.A. Skornyakov,   A.G. Tuganbaev,   "Modules"  ''J. Soviet Math.'' , '''23''' :  6  (1983)  pp. 2642–2706  ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''19'''  (1981)  pp. 31–134</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.R. Goodearl,   "Von Neumann regular rings" , Pitman  (1979)</TD></TR></table>