generated subringGeneratedSubring2007-04-20 16:07:272007-05-03 01:10:06Definitionsubringsubring generated bymonomials in rings% this is the default PlanetMath preamble. as your knowledge
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\begin{defn*} 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 \em{subring generated by} $M$ and is denoted by $\genby{M}$.\end{defn*}
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 $AB$.\newline
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.\newline
The \PMlinkescapetext{composite} of fields is defined only when the respective fields are all included in a larger field.