# Ideal as a projective module

I am sorry, this may not be a good question here. I am looking a good reference about when the ideal $I$ of a given commutative ring $R$ (local or may not be local) with identity is a projective module.

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Here are some partial results if you strengthen the hypotheses on the ring:

Definition: Let $R$ be a domain, $Q=\operatorname{Frac}(R)$. An ideal $I$ is invertible if there are elements $a_1,\dots,a_n\in I$ and $q_1,\dots,q_n \in Q$ such that $q_iI\subset R$ for all $i$, and $1=\sum_{i=1}^n q_ia_i$.

Theorem: If $R$ is a domain, then a nonzero ideal $I$ is projective iff it is invertible.

If we move from domains to UFD's, then:

Theorem: If $R$ is a UFD, then a nonzero ideal $I$ is projective iff it is principal.

A reference for the above theorems is Rotman's "An Introduction to Homological Algebra", 2nd. edition, pp. 167-168.

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Thank you Bruno. –  Rajnish Apr 15 '12 at 18:18
See also Dedekind domains and hereditary rings for (commutative) rings having the property that all ideals are projective. –  t.b. Apr 15 '12 at 18:41
@Rajnish: you're welcome :) –  Bruno Stonek Apr 15 '12 at 20:18