Answer to Question #16738 in Algebra for sanches

Question

Let a, b, c be such that ab + c = 1 in a ring R. If there exists x ∈ R such that a + cx ∈ U(R), show that there exists y ∈ R such that b + yc ∈ U(R).

Answer 1

Write u = a + cx ∈ U(R). We claim that the element y : = (1 − bx)u^−1 works, i.e. v : = b + (1 − bx)u^−1c ∈ U(R). To see this, note that
vx = bx + (1 − bx)u^−1(u − a) = 1 − (1 − bx)u^−1a,
vx(1 − ba) = 1 − ba − (1 − bx)u^−1a(1 − ba) = 1− ba − (1 − bx)u^−1(1 − ab)a = 1− [b + (1 − bx)u^−1c]a = 1− va.
Therefore, for w : = a + x(1 − ba), we have vw = 1. We finish by showing that wv = 1 (for then v ∈ U(R)). Note that
wb = ab + xb(1 − ab) = ab + xbc,
w(1 − bx) = a + x(1 − ba) − abx − xbcx
= a + (1 − ab)x − xb(a + cx)
= a + cx − xbu,
w(1 − bx)u^−1c = c − xbc.
Adding the first and the last equation yields wv = ab + c = 1, as desired.