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generated subring
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(Definition)
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Definition 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 $\genby{M}$ .
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.
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"generated subring" is owned by polarbear.
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Cross-references: complete lattice, generated by, subfield, words, composite field, quotient field, elements, quotients, fields, monomials, sums, finite, subrings, intersection, ring, subset
This is version 6 of generated subring, born on 2007-04-20, modified 2007-05-03.
Object id is 9227, canonical name is GeneratedSubring.
Accessed 1756 times total.
Classification:
| AMS MSC: | 13-00 (Commutative rings and algebras :: General reference works ) | | | 16-00 (Associative rings and algebras :: General reference works ) | | | 20-00 (Group theory and generalizations :: General reference works ) |
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Pending Errata and Addenda
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