What is a projective ideal?

Solution 1:

I don't know of any other meaning of a projective ideal other than the one suggested by Boris Novikov, i.e. an ideal of a ring $R$ that is also projective as an $R$-module. I want to emphasize that such an ideal $I$ need NOT be a direct summand of $R$ (Boris never implied that condition to be necessary - only sufficient!) as well as give more examples.

The obvious examples of that are principal ideals of a commutative domain - they are projective by virtue of being free of rank one.

Another set of projective ideals that comes to mind are the ideals of rings of integers of a number field. If $K$ is a finite extension of $\mathbb{Q}$, and $R={\cal O}_K$ its ring of integers, then every ideal of $R$ is projective. This is because the ring is a Dedekind domain. When the ideal is not principal, it is not isomorphic to $R$ as a module. For example, if $R=\mathbb{Z}[\sqrt{-5}]$ is the ring of integers of the field $\mathbb{Q}[\sqrt{-5}]$, then the prime ideal $P$ generated by the elements $2$ and $1+\sqrt{-5}$ is not isomorphic to $R$ itself or a direct summand of it ($R$ is indecomposable as an $R$-module). However, $P$ can be viewed as a submodule of $R^2$ in such a way that it becomes a direct summand.

Namely, it is easy to verify that for all $x\in P$ we have $x(1-\sqrt{-5})/2\in R$. Therefore we have a well-defined monomorphism $s: P\to R^2, x\mapsto (-x,x(1-\sqrt{-5})/2)$. It is easy to check that this splits the obvious epimorphism $p:R^2\to P, (r_1,r_2)\mapsto 2r_1+(1+\sqrt{-5}) r_2$ by another banal calculation.

Yet another set of examples consists of the Wedderburn components of a complex group algebra $R=\mathbb{C}[G]$ of a finite group $G$. These are the isotypic components of the (left) regular module, and serve as examples of Boris' more general scheme.

Solution 2:

A (left) projective ideal is just a projective submodule of the regular module ${}_RR$ (i.e. the ring $R$ considered as a left $R$-module). So if ${}_RR$ can be decomposed in a direct sum, then every summand is a projective ideal.