In 1999 Nicholson introduced the definition that an element of a ring is called strongly clean if it can be written as the sum of a unit and an idempotent that commute. Similarly, a ring is called strongly clean if each of its elements is strongly clean. While many popular collections of rings have been shown to possess this characteristic, there are some that do not. Perhaps most surprising is the fact that there are still large collections that have yet to be classified. One such example in this final group is the set of formal power series rings. We know not all these structures are strongly clean, but some are. Which ones?
To answer this question we investigated conditions on a ring that imply the extension to a formal power series ring would still be strongly clean. Using Peirce Decompositions and Corners, we were able to isolate the structure needed. Simply stated as a surjective group homomorphism or the solvability of a ring commutator, it is shown that our definition of optimally clean is sufficient to satisfy the extension in question. Further, we were able to to verify the equivalence of strongly and optimally clean within the context of formal power series rings.
Extending on this success, we then investigated similar conditions for an extension to the skew power series ring to be strongly clean. This led to our analogous definition of skew optimally clean and proof of its sufficiency for this result. Additionally, we provide examples verifying our conditions to be distinct from previously established definitions.
Finally, we verify our new conditions to include all other classes that have been shown sufficient to extend to a strongly clean formal power series ring, making this the most general result to date. Unfortunately, there are not yet enough examples to allow us to determine whether or not they are also necessary properties.