How to define a free module on a subring of a ring?

Question

Let $R$ be an associative ring and $A_i$ be a family of subring. Differential extension of $R$ is denoted by $\tilde{R}=\langle R,A_i,t_i,\theta_i, \delta_i: i\in I \rangle$, where $\theta_i: A_i \to R$ is injective homomorphism and $\delta_i: A_i \to R$ is a $\theta_i$-derivation. This extension is defined by universal property. I want to show that $R$ embeds into $\tilde{R}$. For this goal, we need to show that $R/A$ is a free left $A$-module? The question here is how to define a free basis for $R/A_i$?