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Homesemihereditary ring
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semihereditary ring
Let $R$ be a ring. A right (left) $R$module $M$ is called right (left) semihereditary if every finitely generated submodule of $M$ is projective over $R$.
A ring $R$ is said to be a right (left) semihereditary ring if all of its finitely generated right (left) ideals are projective as modules over $R$. If $R$ is both left and right semihereditary, then $R$ is simply called a semihereditary ring.
Remarks.

A hereditary ring is clearly semihereditary.

A ring that is left (right) semiheridtary is not necessarily right (left) semihereditary.

If $R$ is hereditary, then every finitely generated submodule of a free $R$modules is a projective module.

A semihereditary integral domain is a Prüfer domain, and conversely.
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semihereditary module
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minor
Hi, I think
"A ring that is left (right) semihereditary does not mean it is right (left) semihereditary."
should gramatically be
"A ring that is left (right) semiheridtary is not necessarily right (left) semihereditary."
Also, I think you left off a "(left)" in the first line after "finitely generated."
Thanks,
Cam
Re: minor
Whoops, that was supposed to be a correction, not a post.
Cam