The subring generated by $M$ is formed by finite sums of monomials of the form :

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:

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.