Let's say we have a BGV style homomorphic encryption scheme. The message space will be the ring $$R_p = \mathbb Z_p[x]/(x^d + 1)$$ where $p$ is a prime congruent to $1$ modulo $2d$. Now let's say we say messages $m_1(x), m_2(x) \in R_p.$ How do we obtain a ciphertext encrypting both $m_1(x)$ and $m_2(x)$? The BGV paper mentions the CRT isomorphism $$R_p \cong R_{\mathscr{p_1}} \times ... \times R_{\mathscr{p_d}}.$$ Under this isomorphism, we have the mapping $m_1(x) \to ((m_{1,1})(x),...,(m_{1,d})(x))$ and we have a similar representation for $m_2(x)$. I'm still not sure how we use this mapping to get a ciphertext encrypting both $m_1(x)$ and $m_2(x)$ at the same time however.

Any clarification would be greatly appreciated.