課程名稱︰代數導論一
課程性質︰系定必修
課程教師︰莊正良 教授
開課系所︰數學系
考試時間︰2005.1.15
試題： ┌────────────────┐
│Ｚ: the ring of integers │
│Ｑ: the ring of rationals │
│Ｃ: the ring of complex numbers │
└────────────────┘
1.(15%) Are rings 2Ｚ and 4Ｚ isomorphic?
2.(15%) Are rings Ｑ[√2] and Ｑ[√3] isomorphic?
3.(15%) Let the ring homomorphism φ:Ｑ[x] → Ｑ[√7 + i] be defined by
φ(a) = a for a∈Q and φ(x) = √7 + i. Find the kernel and the image
of φ. Hint: Ｑ[x] is a PID. Work in Ｃ[x] first.
(註：√7 + i 中的 i 指的是虛數)
4.(15%) Show that Ｚ[√-2] is a Euclidean domain.
5.(15%) (a) State Zorn's Lemma. (b) Let R be a ring with 1. Prove that for
any ideal I of R, if I≠R, then I can be extended to a maximal ideal of R.
6.(15%) Let R be a commutative ring. Show that the set of nilpotent elements
of R forms an ideal N and that the quotient ring R/N does not possess
nonzero nilpotent elements.
7.(15%) Let R be a PID and I a nonzero ideal of R. Show that R/I has only
finitely many ideals.
8.(15%) Let R be a commutative ring with 1. Given a∈R, let I be the ideal
of R[x] generated by 1-ax. Show that I∩R = 0 iff for any b∈R, ab = 0
implies b = 0.
9.(15%) Let a, R, I be as given in Problem 8. Let φ: R → S be a ring
homomorphism of R into another commutative ring S with 1 such that φ(a)
is invertible in S. Show that there is a ring homomorphism
ψ: R[x]/I → S such that ψ(r+I) = ψ(r) for r∈R, where r+I denotes
the equivalence class of r in the quotient ring R[x]/I.
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