hereditary ring

Let $R$ be a ring. A right (left) $R$ -module $M$ is called right (left) hereditary if every submodule of $M$ is projective over $R$ .

Remarks.

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If $M$ is semisimple, then $M$ is hereditary.
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Suppose $M$ is an external direct sum of hereditary right (left) $R$ -modules, then $M$ is itself hereditary.

A ring $R$ is said to be a right (left) hereditary ring if all of its right (left) ideals are projective as modules over $R$ . If $R$ is both left and right hereditary, then $R$ is simply called a hereditary ring.

Remarks.

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Even though the notions of left and right heredity in rings are symmetrical, one does not imply the other.
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If $R$ is semisimple, then $R$ is hereditary.
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If $R$ is hereditary, then every free $R$ -module is a hereditary module.
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A hereditary integral domain is a Dedekind domain, and conversely.
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The global dimension of a non-semisimple hereditary ring is 1.

单词	HereditaryRing
释义	hereditary ring Let $R$ be a ring. A right (left) $R$ -module $M$ is called right (left) hereditary if every submodule of $M$ is projective over $R$ . Remarks. • If $M$ is semisimple, then $M$ is hereditary. • Suppose $M$ is an external direct sum of hereditary right (left) $R$ -modules, then $M$ is itself hereditary. A ring $R$ is said to be a right (left) hereditary ring if all of its right (left) ideals are projective as modules over $R$ . If $R$ is both left and right hereditary, then $R$ is simply called a hereditary ring. Remarks. • Even though the notions of left and right heredity in rings are symmetrical, one does not imply the other. • If $R$ is semisimple, then $R$ is hereditary. • If $R$ is hereditary, then every free $R$ -module is a hereditary module. • A hereditary integral domain is a Dedekind domain, and conversely. • The global dimension of a non-semisimple hereditary ring is 1.
随便看	LocalFinitenessIsClosedUnderExtensionProofThat local finiteness is closed under extension, proof that LocalHomeomorphism local homeomorphism LocalHomeomorphismsBetweenRealNumbers local homeomorphisms between real numbers Localization localization LocalizationForDistributions localization for distributions LocalizationOfAModule localization of a module LocalizationsOfDedekindDomainsAreDedekind localizations of Dedekind domains are Dedekind LocallyBounded locally bounded LocallycalP LocallyClosed locally closed LocallyClosedSubgroupsOfTopologicalGroupsAreClosed locally closed subgroups of topological groups are closed LocallyCompact locally compact LocallyCompactGroupoids locally compact groupoids