Let $\mathcal{L}$ be a first-order language, and $M$ and $N$ be two $\mathcal{L}$-structures. We say that $M$ and $N$ are elementarily equivalent (write $M \approx N$) if they satisfy the same first-order formulas in the language $\mathcal{L}$. It is clear that if $M$ and $N$ are isomorphic (write $M \simeq N$), then $M \approx N$, but the converse does not holds (see below).
It is an important problem in algebra, given an algebraic structure $M$, to determine what algebraic properties are determined by its elementary equivalence class. This is related to the problem of determining which algebraic properties can be expressed in first-order logic, and has some classical problems, such as Tarski's Problem As a trivial exemple, given two fields $K$ and $L$, if $K \approx L$, then they have the same characteristic; as a less trivial example, if $K \approx L$ then $K$ is algebraically closed if and only iff $L$ is algebraically closed.
As an example of two elementarily equivalent structures which are not isomorphic, we can consider the algebraic closures $K$ and $L$ of $\mathbb{Q}(x)$ and $\mathbb{Q}(x,y)$, respectively. Since the theory of algebraically closed fields of zero characteristic is complete, clearly $K \approx L$; however, the fields are not isomorphic, as $K$ has transcendece degree $1$ over its prime field; and $L$ has transcendence degree $2$.
The theory of elementary equivalence of fields is a mature subject with many references (for one which I found very interesting, we have this)
In looking for references for the theory of elementary equivalence of rings. I know some classical results (for example, if the rings $R$ and $S$ are elementarily equivalent, $R$ is Jacobson-semisimple if and only if $S$ is Jacobson-semisimple; if $R$ and $S$ are elementarily equivalent commutative Noetherian rings and $R$ has finite Krull dimension $n$, then $S$ also has Krull dimension $n$; the first-order theory of rings is undecidable; etc), but I could not find many papers dealing specifically with this topic.
An exception is the very interesting paper by Kharlampovich and Myasnikov about the analogue of Tarski's Problem for the free associative algebras (the paper can be found here).
So, I'm interested in surveys and important papers about the elementary theory of rings.
