Let $\Phi_S$ be an operator acting on a space $\mathcal H_S$. If we introduce an ancilla $A$, the total space becomes $\mathcal H_S\otimes \mathcal H_A$ and I can naturally extend the operator $\Phi_S$ to act on the whole space by defining $\Phi_{SA}=\Phi_S\otimes \mathbb{I}_A$, where $\mathbb{I}_A$ is the identity on $A$.
What is the correct way of extending a state or more generally a density operator $\rho_S$ to the space $\mathcal H_S\otimes \mathcal H_A$? Since $\rho_S$ is also an operator, I would write again $\rho_{SA}=\rho_S\otimes \mathbb I_A$. Is this correct?