Solutions of $f(x)=f(a)+(RL)\int _{a}^{x} (fH+fG)$ for rings

We show that there is a solution $f$ of the equation \[ f(x) = f(a) + (RL)\int _a^x {(fH + fG)} \] such that $f(p) = 0$ and $f(q) \ne 0$ for some pair $p,q \in [a,b]$ iff there is a number $t \in [a,b]$ such that one of $1 - H({t^ - },t),1 - H(t,{t^ + }),1 + G({t^ - },t)$ or $1 + G(t,{t^ + })$ is zero or a right divisor of zero, where $f,G$ and $H$ are functions of bounded variation with ranges in a normed ring $N$. Furthermore, if $N$ is a field, then for each discontinuity of $H$ on $[a,b]$ there exists $\lambda \in N$ and a finite set of linearly independent nonzero solutions on $[a,b]$ of the equation $f(x) = f(a) + (RL)\int _a^x {(fH + fG)\lambda }$ such that if $f$ is a solution and has bounded variation on $[a,b]$, then $f$ is a linear combination of this set of solutions. Product integrals are used extensively in the proofs.