College Algebra Exam Review 258

College Algebra Exam Review 258 - 268 6 RINGS Instead of...

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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 coefﬁcient-by-coefﬁcient, 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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