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

In summary: An ideal of an algebra defines a module, but the converse is not always true. However, when modules are theta-stable and invariant under the U(1)^2 symmetry, there is a one-to-one correspondence between ideals and modules. This is also known as a one-to-one correspondence between isomorphism classes of indecomposable M-injective modules and prime M-ideals in a left R-module M.
  • #1
Monocles
466
1
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" [Broken] page 29:

Generally speaking, an ideal of an algebra defines a module. To see this, consider a vector [itex]|\mathcal{I} \rangle[/itex] which is annihilated by all elements of the ideal [itex]\mathcal{I}[/itex]. From [itex]|\mathcal{I}\rangle[/itex], we can generate a finite-dimensional representation of the algebra [itex]A[/itex] by acting with elements of [itex]A[/itex] on it. However, the converse is not always true. Fortunately, when modules are [itex]\hat{\theta}[/itex]-stable and invariant under the [itex]U(1)^2[/itex] 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?
 
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  • #2
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!
 
  • #3
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
 

1. What is the definition of one-to-one correspondence between ideals and modules of an algebra?

One-to-one correspondence between ideals and modules of an algebra refers to a relationship where each ideal of the algebra is uniquely associated with a module, and vice versa. This means that for every ideal, there is exactly one corresponding module, and for every module, there is exactly one corresponding ideal.

2. How does one-to-one correspondence between ideals and modules benefit algebraic structures?

One-to-one correspondence between ideals and modules is important because it allows for a better understanding of the structure of an algebraic system. It helps to establish a clear relationship between the two concepts and allows for the translation of properties and operations from one to the other. This can aid in solving algebraic problems and proving theorems.

3. Can a module have multiple corresponding ideals?

No, a module can only have one corresponding ideal. This is because one-to-one correspondence means that each element in one set has a unique element in the other set. Therefore, a module cannot be associated with multiple ideals, as it would violate the one-to-one correspondence relationship.

4. How is one-to-one correspondence between ideals and modules related to isomorphism?

One-to-one correspondence between ideals and modules is closely related to the concept of isomorphism. If there is a one-to-one correspondence between the ideals and modules of an algebra, then the algebra is said to be isomorphic to itself. This means that the two structures have the same underlying properties and can be considered equivalent.

5. Can one-to-one correspondence between ideals and modules be extended to other algebraic structures?

Yes, one-to-one correspondence between ideals and modules can be extended to other algebraic structures such as rings and fields. This is because the concept of one-to-one correspondence is a general mathematical concept that can be applied to any two sets with a well-defined relationship. However, the specific properties and operations of each structure may differ.

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