In this paper, we introduce the new notion of n-f-clean rings as a generalization of f-clean rings. Next, we investigate some properties of such rings. We prove that $M_n(R)$ is n-f-clean for any n-f-clean ring R. We also, get a condition under which the denitions of n-cleanness and n-f-cleanness are equivalent.

We introduce the tropical $F$-polynomial $f_M$ of a quiver representation $M$. study its interplay with general presentation for any finite-dimensional basic algebra. give an interpretation evaluating at weight vector. As consequence, we Newton polytope ${\sf N}(M)$ dual fan and 1-skeleton N}(M)$. propose algorithm to determine generic polytopes, show it works path algebras. application, repres...