generated subring

The subring generated by $M$ is formed by finite sums of monomials of the form :\begin{equation*} a_1a_2 \cdots a_n, \mbox{where} \;\;\displaystyle a_1,\ldots , a_n \in M.\end{equation*}Of particular interest is the subring generated by a family of subrings $E = \{A_i|\;\; i\in I\}$ . It is the ring $R$ formed by finite sums of monomials of the form:\begin{equation*} \displaystyle a_{i_1}a_{i_2} \ldots a_{i_n}, \mbox{where}\;\; a_{i_k} \in A_{i_k}. \end{equation*}If $A,B$ are rings, the subring generated by $A \cup B$ is also denoted by $AB$ .
In the case when $A_i$ are fields included in a larger field $A$ then the set of all quotients of elements of $R$ ( the quotient field of $R$ ) is the composite field $\bigvee_{i\in I}A_i$ of the family $E$ . In other words, it is the subfield generated by $\bigcup_{i\in I}A_i$ . The notation $\bigvee_{i\in I}A_i$ comes from the fact that the family of all subfields of a field forms a complete lattice.
The composite of fields is defined only when the respective fields are all included in a larger field.