College Algebra Exam Review 269

College Algebra Exam Review 269 - 6.2 HOMOMORPHISMS AND...

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Unformatted text preview: 6.2. HOMOMORPHISMS AND IDEALS 279 Ideals in Z and in KŒx In the ring of integers, and in the ring KŒx of polynomials in one variable over a field, every ideal is principal: Proposition 6.2.27. (a) For a subset S  Z, the following are equivalent: (i) S is a subgroup of Z. (ii) S is a subring of Z. (iii) S is an ideal of Z. (b) Every ideal in the ring of integers is principal. (c) Every ideal in KŒx, where K is a field, is principal. Proof. Clearly an ideal is always a subring, and a subring is always a subgroup. If S is a nonzero subgroup of Z, then S D Zd , where d is the least positive element of S , according to Proposition 2.2.21 on page 98. If S D f0g, then S D Z0. In either case, S is a principal ideal of Z. This proves (a) and (b). The proof of (c) is similar to that of Proposition 2.2.21. The zero ideal of KŒx is clearly principal. Let J be a nonzero ideal, and let f 2 J be a nonzero element of least degree in J . If g 2 J , write g D qf C r , where q 2 KŒx, and deg.r/ < deg.f /. Then r D g qf 2 J . Since deg.r/ < deg.f / and f was a nonzero element of least degree in J , it follows that r D 0. Thus g D qf 2 KŒxf . Since g was an arbitrary element of J , J D KŒxf . I Direct Sums Consider a direct sum of rings R D R1 ˚ ˚ Rn . For each i , set Q Q Ri D f0g ˚ ˚ f0g ˚ Ri ˚ f0g ˚ ˚ f0g. Then Ri is an ideal of R. How can we recognize that a ring R is isomorphic to the direct sum of several subrings A1 ; A2 ; : : : ; An ? On the one hand, according to the previous example, the component subrings must actually be ideals. On the other hand, the ring must be isomorphic to the direct product of the Ai , regarded as abelian groups. These conditions suffice. Proposition 6.2.28. Let R be a ring with ideals A1 ; : : : As such that R D A1 C C As . Then the following conditions are equivalent: (a) .a1 ; : : : ; as / 7! a1 C C as is a group isomorphism of A1 As onto R. (b) .a1 ; : : : ; as / 7! a1 C C as is a ring isomorphism of A1 ˚ ˚ As onto R. ...
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