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 ${\left\langle M\right\rangle}$.

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

 $a_{1}a_{2}\cdots a_{n},\mbox{where}\;\;\displaystyle a_{1},\ldots,a_{n}\in M.$

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:

 $\displaystyle a_{i_{1}}a_{i_{2}}\ldots a_{i_{n}},\mbox{where}\;\;a_{i_{k}}\in A% _{i_{k}}.$

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.

Title generated subring GeneratedSubring 2013-03-22 16:57:27 2013-03-22 16:57:27 polarbear (3475) polarbear (3475) 9 polarbear (3475) Definition msc 20-00 msc 13-00 msc 16-00 RingAdjunction subring generated by monomials in rings