Let $P$ and $B$ be two unital rings and let $f\colon P\to R$ be a unital injective ring homomorphism. Suppose that $a_1, \ldots, a_n \in P$ are elements such that
$R=R\cdot f(a_1) + \ldots + R\cdot f(a_n)$.
Do we have
$P = P\cdot a_1 + \ldots + P \cdot a_n$ ?