Today, I was on and off wondering about idempotents in **Z**-graded associative algebras with a unit (which is assumed homogeneous). In my googling of this, I have found the terminology *graded idempotents* which refers to idempotents which are homogeneous of degree 1. This suggests that there exist others. And indeed, it is easy to make examples of non-homogeneous idempotents by conjugation with units. But we can ask for more.

- Is there an example of an idempotent which is not conjugate to a graded idempotent?
- Is there an example of a
**Z**-graded associative algebra with a nontrivial idempotent but no nontrivial graded idempotents?

Hmm…?

Some more searching and google finally turned up the paper *Idempotents in ring extensions* by Kanwar, Leroy, and Matczuk which provides the answer to 1. There’s probably tons of papers that make this observation. Namely, suppose that R is a (commutative) domain such that R[x, x^{-1}] and R don’t have the same Picard group. For example R = k[t^2, t^3] with k a field (details omitted). Let L be an invertible module over R[x, x^{-1}] which is not isomorphic to the pullback of an invertible module from R. Pick a surjection

R[x, x^{-1}]

^{⊕ n}—> L

As L is a projective R[x, x^{-1}]-module we obtain an idempotent e in the **Z**-graded ring M_n(R[x, x^{-1}]) = M_n(R)[x, x^{-1}]. And this idempotent is not conjugate to an element of M_n(R) as that would mean L does come from R.

So this answers 1. I do not know the answer to 2.