Word for one-to-one correspondence between ideals and modules of an algebra

[Monocles]

I do not know if this is a common/standard construction, so here is my motivation for this question. From http://arxiv.org/abs/1002.1709" page 29:

Generally speaking, an ideal of an algebra defines a module. To see this, consider a vector |\mathcal{I} \rangle which is annihilated by all elements of the ideal \mathcal{I}. From |\mathcal{I}\rangle, we can generate a finite-dimensional representation of the algebra A by acting with elements of A on it. However, the converse is not always true. Fortunately, when modules are \hat{\theta}-stable and invariant under the U(1)^2 symmetry, it was shown in Refs 33, 34 that there is a one-to-one correspondence between ideals and modules.

Is there a word for when there is such a one-to-one correspondence?

 

[Monocles]

After rereading this today, I think he might just be saying there is a one-to-one correspondence between ideals and theta-stable U(1)^2-invariant A-modules specifically, not ALL A-modules. So this question probably does not have an answer!

 

[kdbnlin78]

For a left R-module M, you can identify certain submodules of M that are similar to that of prime ideals in a ring, R. With that definition there exists conditions on the module M which imply that there is a one-to-one correspondence between isomorphism classes of indecomposable M-injective modules and prime M-ideals.

Hope that helps.
Regards,
kdbnlin