Let R be a commutative ring with identity and G be a finite group suchthat G{91, 92, ..., In}. Consider thei=nRIG] = {a = > a;gi : a; E R}.i=1Then R[G] is a ring with respect to the following operations:For a = E a;9g; and 3 = EE b;g; € R[G],vi=1i=na + ß = ) (a; + b;)g;i=1andi=naß = DCk Jk,k=1where c = Ea,:=0, a;b;. If I is an ideal of R show that I[G] is an idealof R[G], wheregi9j=9ki=nI[G] = {a = > a;g; : a; E I}.i=1

Definition of an Ideal: Let R, +, · be a ring, a subset I of R is said to be an ideal of R, if, I,…

Let R be a commutative ring with identity and G be a finite group such that G = {g1, g2 ..., In}. Consider the i=n R[G] = {a = > a¿gi : a¡ E R}. %3D i=1 Then R[G] is a ring with respect to the following operations: C 4:9; and 3 =EET b:9; € R[G], i=n For a = i=n a + ß = > (a; + b;)g; i=1 and i=n aß = > Ck9k; k=1 where c = Eo:9:=0 a;b;. If I is an ideal of R show that I[G] is an ideal of R[G], where (gi9j=9k i=n I[G] = {a = > a;g; : a; E I}. i=1

Let R be a commutative ring with identity and G be a finite group such that G = {g1, g2 ..., In}. Consider the i=n R[G] = {a = > a¿gi : a¡ E R}. %3D i=1 Then R[G] is a ring with respect to the following operations: C 4:9; and 3 =EET b:9; € R[G], i=n For a = i=n a + ß = > (a; + b;)g; i=1 and i=n aß = > Ck9k; k=1 where c = Eo:9:=0 a;b;. If I is an ideal of R show that I[G] is an ideal of R[G], where (gi9j=9k i=n I[G] = {a = > a;g; : a; E I}. i=1

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let R be a commutative ring with identity and G be a finite group such
that G
{91, 92, ..., In}. Consider the
i=n
RIG] = {a = > a;gi : a; E R}.
i=1
Then R[G] is a ring with respect to the following operations:
For a = E a;9g; and 3 = EE b;g; € R[G],
vi=1
i=n
a + ß = ) (a; + b;)g;
i=1
and
i=n
aß = D
Ck Jk,
k=1
where c = Ea,:=0, a;b;. If I is an ideal of R show that I[G] is an ideal
of R[G], where
gi9j=9k
i=n
I[G] = {a = > a;g; : a; E I}.
i=1

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