Semi-hereditary ring

from the left

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 is a semi-hereditary ring and there is an with , then 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 , the following properties are equivalent: 1) is semi-hereditary; 2) , where , and are arbitrary ideals in ; 3) the complete ring of fractions of is regular in the sense of von Neumann, and for every maximal ideal of the ring of fractions is a normal ring; and 4) all -generated ideals of are projective. The ring of polynomials in one variable over a commutative ring is semi-hereditary if and only if is regular in the sense of von Neumann.

References

[1]	H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[2a]	L.A. Skornyakov, A.V. Mikhalev, "Modules" Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 14 (1976) pp. 57–190 (In Russian)
[2b]	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

References

[a1]	K.R. Goodearl, "Von Neumann regular rings" , Pitman (1979)