Exercise 21, § 1

Let A be a ring, $b, c$ two elements of A which are not right divisors of 0, M the A-module $A/Abc$ and N the submodule $Ac/Abc$ of M. For N to be a direct factor in M, it is necessary and sufficient that there exist two elements $x, y$ of A such that $xb + cy = 1$. (If $\varepsilon$ is the class of 1 in M, show that a supplementary subspace of N in M is generated by an element of the form $(1 - yc)\varepsilon$ whose annihilator is the ideal $Ac$.)