Right ideals generated by an idempotent of finite rank

Let R be a K-algebra acting densely on VD, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let f(X1,…,Xt) be an arbitrary and fixed polynomial over K in noncommuting indeterminates X1,…,Xt with constant term 0 such that for some μ∈K occurring in the coefficients of f(X1,…,Xt). It is proved that a right ideal ρ of R is generated by an idempotent of finite rank if and only if the rank of f(x1,…,xt) is bounded above by a same natural number for all x1,…,xt∈ρ. In this case, the rank of the idempotent that generates ρ is also explicitly given. The results are then applied to considering the triangularization of ρ and the irreducibility of f(ρ), where f(ρ) denotes the additive subgroup of R generated by the elements f(x1,…,xt) for x1,…,xt∈ρ.