generated subring

Definition 1

Let $M$ be a nonempty subset of a ring $A$. The intersection of all subrings of $A$ that include $M$ is the smallest subring of $A$ that includes $M$. It is called the subring generated by $M$ and is denoted by $\mathrm{⟨}M\mathrm{⟩}$.

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

$$a_1{a}_2\mathrm{\cdots }{a}_n,\text{where}{a}_1,\mathrm{\dots },a_n\in M.$$

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:

$$a_{i_1}{a}_{i_2}\mathrm{\dots }{a}_{i_n},\text{where}{a}_{i_k}\in A_{i_k}.$$

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 of fields is defined only when the respective fields are all included in a larger field.