Difference between revisions of "Semi-hereditary ring"
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| − | 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 | + | 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 $R$ 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 | + | 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. |
====References==== | ====References==== | ||
Latest revision as of 17:47, 27 August 2014
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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.
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 |
Comments
References
| [a1] | K.R. Goodearl, "Von Neumann regular rings" , Pitman (1979) |
How to Cite This Entry:
Semi-hereditary ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-hereditary_ring&oldid=33177
Semi-hereditary ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-hereditary_ring&oldid=33177
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article