Exercise 3, § 5

Let B be a ring and I a two-sided ideal of B; let A be the subring of $B\times B$ consisting of the pairs $(x, y)$ such that $x-y\in I$. Let $B_1$ (resp. $B_2$) be the additive group B endowed with an A-module structure deduced from the first (resp. second) projection. Prove that the A-modules $B_1$ and $B_2$ are balanced. Under what condition on I is the A-module $B_1\oplus B_2$ balanced?