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Unformatted text preview: 268 6. RINGS Instead of taking coefﬁcients in Z, we can also take coefﬁcients in C ,
for example; the result is called the complex group ring of G .
Example 6.1.9. Let R be a commutative ring with multiplicative identity
element. A formal power series in one variable with coefﬁcients in R is a
P
formal inﬁnite sum 1 0 ˛i x i . The set of formal power series is denoted
iD
RŒŒx. Formal power series are added coefﬁcientbycoefﬁcient,
1
1
1
X
X
X
˛i x i C
ˇi x i D
.˛i C ˇi /x i :
i D0 i D0 i D0 The product of formal power series is deﬁned as for polynomials:
1
1
1
X
X
X
i
i
i
.
˛i x /.
ˇi x / D
ix ;
i D0 i D0 i D0 Pn where n D j D0 ˛j ˇn j . With these operations, the set of formal power
series is a commutative ring. Deﬁnition 6.1.10. Two rings R and S are isomorphic if there is a bijection
between them that preserves both the additive and multiplicative structures.
That is, there is a bijection ' W R ! S satisfying '.a C b/ D '.a/ C '.b/
and '.ab/ D '.a/'.b/ for all a; b 2 R. Deﬁnition 6.1.11. The direct sum of several rings R1 ; R2 ; : : : ; Rn is the
Cartesian product endowed with the operations
0
0
0
0
0
0
.r1 ; r2 ; : : : ; rn / C .r1 ; r2 ; : : : ; rs / D .r1 C r1 ; r2 C r2 ; : : : ; rn C rn / and 0
0
0
0
0
0
.r1 ; r2 ; : : : ; rn /.r1 ; r2 ; : : : ; rs / D .r1 r1 ; r2 r2 ; : : : ; rn rn /:
The direct sum of R1 ; R2 ; : : : ; Rn is denoted R1 ˚ R2 ˚ ˚ Rn . Example 6.1.12. According to the discussion at the end of Section 1.11
and in Exercise 1.11.10, if a and b are relatively prime natural numbers,
then Zab and Za ˚Zb are isomorphic rings. Consequently, if a1 ; a2 ; : : : ; an
are pairwise relatively prime, then Za1 a2 an Š Za1 ˚ Za2 ˚
˚ Zan
as rings. ...
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 Fall '08
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 Algebra, Power Series

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