Endomorphisms That Are the Sum of a Unit and a Root of a Fixed Polynomial

摘要

If C = C(R) denotes the center of a ring R and g(x) is a polynomial in C[x], Camillo and Simón called a ring g(x)-clean if every element is the sum of a unit and a root of g(x). If V is a vector space of countable dimension over a division ring D, they showed that end DV is g(x)-clean provided that g(x) has two roots in C(D). If g(x) = x - x2 this shows that end DV is clean, a result of Nicholson and Varadarajan. In this paper we remove the countable condition, and in fact prove that Mend RM is g(x)-clean for any semisimple module M over an arbitrary ring R provided that g(x) in (x - a)(x - b)C[x] where a, b in C and both b and b - a are units in R.