Filtrations in semisimple rings

In this paper, we describe the maximal bounded $\mathbb{Z}$-filtrations of Artinian semisimple rings. These turn out to be the filtrations associated to finite $\mathbb{Z}$-gradings. We also consider simple Artinian rings with involution, in characteristic $\neq 2$, and we determine those bounded $\mathbb{Z}$-filtrations that are maximal subject to being stable under the action of the involution. Finally, we briefly discuss the analogous questions for filtrations with respect to other Archimedean ordered groups.