I'm reading Model theory: an introduction, by David Marker. I'm at page 14, where it says:
...one way to get a theory is to take $\operatorname{Th}(\mathcal{M})$, the full theory of an $\mathcal{L}$-structure $\mathcal{M}$. In this case, the elementary class of models of $\operatorname{Th}(\mathcal{M})$ is exactly the class of $\mathcal{L}$-structures elementarily equivalent to $\mathcal{M}$....
I have some problem understanding these few lines. In fact, call $\mathcal{K}$ the class of models of $\operatorname{Th}(\mathcal{M})$. Let $\mathcal{A}\in\mathcal{K}$. Let $\phi$ be a sentence which is true in $\mathcal{M}$. Then $\phi$ is also true in $\mathcal{A}$. But why should the converse also hold? In order to have $\mathcal{A}\equiv\mathcal{M}$ (i.e. $\mathcal{A}$ and $\mathcal{M}$ elementary equivalent) each sentence which is true in $\mathcal{A}$ must be true also in $\mathcal{M}$. By construction we have: true in $\mathcal{M}$ implies true in $\mathcal{A}$, but the opposite implication sounds strange to me. For instance, let $\mathcal{M}$ be a group, and $\mathcal{A}$ a ring. Then every sentence $\phi$ in $\operatorname{Th}(\mathcal{M})$ is true in $\mathcal{A}$, but why do every sentence which is true for a ring is also true for a group?
Suppose that $\mathcal{A}\vDash\varphi$. If $\mathcal{M}\not\vDash\varphi$, then $\mathcal{M}\vDash\neg\varphi$, so $\neg\varphi\in\operatorname{Th}(\mathcal{M})$, and $\mathcal{A}\vDash\neg\varphi$. Oops!