Common solutions of linear equations in a ring, with applications

Abstract

This paper gives necessary and sufficient conditions for the existence of a common solution, and two expressions for the general common solution of the equation pair a_1xb_1 = c1, a_2xb_2 = c2, via a simpler equation p_1xp_2 + q_1yq_2=c, when each element belongs to an associative ring with unit. The paper also considers the same problem in the setting of a strongly â-reducing ring. Solutions of the generalized Sylvester equation are also presented. Both the solvability conditions and the expression for the general solution are given in terms of inner inverses. The paper uses the results obtained in the ring setting to give equivalent results for operators between Banach spaces, thus also recovering some of the well known matrix results.