My question come from this
"A structure ${\mathcal {M}}$ is said to be a model of a theory T if the language of $\mathcal {M}$ is the same as the language of T and every sentence in T is satisfied by $\mathcal {M}$. Thus, for example, a "ring" is a structure for the language of rings that satisfies each of the ring axioms, and a model of ZFC set theory is a structure in the language of set theory that satisfies each of the ZFC axioms." https://www.wikiwand.com/en/Structure_(mathematical_logic)
I don't get
"if the language of $\mathcal {M}$ is the same as the language of T"
Then they set an example
"Thus, for example, a "ring" is a structure for the language of rings that satisfies each of the ring axioms"
I don't know what the language of ring is : for me rings is a formal theory constructed over a first order logic language (hence a formal system ), a language which can have different formal symbols (provided by its alphabet) from one of its model : if one takes the set of integer, one can substitutes some formal symbols of the theory with integers and addition symbol and multiplication symbol (if they were not used already in the theory), though the syntax remains, one changed some symbols, therefore the language used by the model is not the same as in the theory, where am I wrong?
Suppose I have a ring $\mathcal{R}=(R; +^\mathcal{R},\times^\mathcal{R}, 0^\mathcal{R},1^\mathcal{R})$. There are two first-order languages which are naturally associated to $\mathcal{R}$:
The "nothing-taken-for-granted" language $\Sigma_{rings}$ which just has the binary function symbols $+$ and $\times$ and the constant symbols $0$ and $1$. This is the "language of rings."
The "all-about-$\mathcal{R}$" language $\Sigma_{\mathcal{R}}$ which has, in addition to the four symbols above, a constant symbol $c_r$ for every $r\in R$.
There's a crucial subtlety here: despite the tight connection between $\mathcal{R}$ and $\Sigma_\mathcal{R}$, the structure $\mathcal{R}$ is not a $\Sigma_\mathcal{R}$-structure but merely a $\Sigma_{rings}$-structure; this was indicated when I wrote $$\mathcal{R}=(R; +^\mathcal{R},\times^\mathcal{R}, 0^\mathcal{R},1^\mathcal{R})$$ at the beginning of this answer. What is true is that $\mathcal{R}$ has a natural expansion $\mathcal{R}_{nameeverything}$ to $\Sigma_\mathcal{R}$. We often conflate $\mathcal{R}$ and $\mathcal{R}_{nameeverything}$ and e.g. write "$\mathcal{R}\models r+s=t$" in place of "$\mathcal{R}_{nameeverything}\models c_r+c_s=c_t$," but strictly speaking they are different objects.