How to pronounce “group ring”

Meaning

• [Noun]
• (algebra) Given ring R with identity not equal to zero, and group
• G
• =
• {
• g
• 1
• ,
• g
• 2
• ,
• .
• .
• .
• ,
• g
• n
• }
• {\displaystyle G=\{g_{1},g_{2},...,g_{n}\}}
• , the group ring RG has elements of the form
• a
• 1
• g
• 1
• +
• a
• 2
• g
• 2
• +
• .
• .
• .
• +
• a
• n
• g
• n
• {\displaystyle a_{1}g_{1}+a_{2}g_{2}+...+a_{n}g_{n}}
• (where
• a
• i
• ∈
• R
• {\displaystyle a_{i}\in R}
• ) such that the sum of
• a
• 1
• g
• 1
• +
• a
• 2
• g
• 2
• +
• .
• .
• .
• +
• a
• n
• g
• n
• {\displaystyle a_{1}g_{1}+a_{2}g_{2}+...+a_{n}g_{n}}
• and
• b
• 1
• g
• 1
• +
• b
• 2
• g
• 2
• +
• .
• .
• .
• +
• b
• n
• g
• n
• {\displaystyle b_{1}g_{1}+b_{2}g_{2}+...+b_{n}g_{n}}
• is
• (
• a
• 1
• +
• b
• 1
• )
• g
• 1
• +
• (
• a
• 2
• +
• b
• 2
• )
• g
• 2
• +
• .
• .
• .
• +
• (
• a
• n
• +
• b
• n
• )
• g
• n
• {\displaystyle (a_{1}+b_{1})g_{1}+(a_{2}+b_{2})g_{2}+...+(a_{n}+b_{n})g_{n}}
• and the product is
• ∑
• k
• =
• 1
• n
• (
• ∑
• g
• i
• g
• j
• =
• g
• k
• a
• i
• b
• j
• )
• g
• k
• {\displaystyle \sum _{k=1}^{n}\left(\sum _{g_{i}g_{j}=g_{k}}a_{i}b_{j}\right)g_{k}}
• .

Practice Nearby Words

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Similar Sounds

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