Angular Momentum weird scenario

[PhysicsKid703]

Homework Statement

Before anything, please address this first question:
Angular momentum, at least for pure rotation, is defined about the axis of rotation only, correct?
[/B]
We are given a wooden ring of mass M, radius R.
I place a small particle of mass M on its circumference.
Now the wooden ring is put into pure rotation.

This is a self made conceptual question so if you're wondering why it's so weird, that's why.
my questions;
a) How do we calculate Angular Momentum of the body? What do we define it about?

Because we know that
--->for a normal rigid body in pure rotation about it's axis of symmetry(which passes through COM), angular momentum is Icm*w , where Icm is moment of inertia about COM and w is angular velocity.
--->for a normal rigid body in pure rotation about any other axis of rotation other than its axis of symmetry(which still passes through COM) , angular momentum is Iaxis*w , where laxis is moment of inertia about that axis and w is again angular velocity.
(All this is obtained through ∫ dm r²w , where r in the first case is position vector of each particle from axis of rotation, and the same in the second case, right?)

The main reason I'm getting confused is that the situation in question lies somewhere in the nether:
The body is in pure rotation about its axis of symmetry, but the axis of symmetry is not passing through the com.

The Attempt at a Solution

My thoughts are this, so correct me if I'm wrong:
I have no thoughts I'm blank.
Please explain elaborately for I have lost almost all patience with angular momentum today.

 

[mfb]

Angular momentum, at least for pure rotation, is defined about the axis of rotation only, correct?

Every axis has a well-defined angular momentum. It does not matter which object rotates around which axis.
The two cases you describe are just special cases - you can still use them to find the contribution of the individual components, and add them.