Download Homework 7

MATH 258 HOMEWORK #7
DUE THURSDAY, FEBRUARY 19
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(2)
(3)
(4)
Do exercise 13 in section 7.2 in Dummit & Foote.
Do exercises 12,13,15,20 in section 7.3 in Dummit & Foote.
Do exercises 6-8,11,12 in section 7.4 in Dummit & Foote.
Let R be a ring. (R, +) is an abelian group, and thus a Z-module. Show that the multiplication in R gives a homomorphism of abelian groups R ⊗Z R → R.
(5) Let R be a ring. Rop is defined to have the same underlying additive structure as R, but
multiplication is done in the opposite direction. Thus the product of a · b in Rop is the
element ba. Show that Rop is a ring, and that a left ideal of Rop is a right ideal of R.
(6) Let R be a commutative ring. A polynomial p(x) ∈ R[x] is called irreducible if whenever
we write p(x) = q(x)r(x) we must have either q(x) or r(x) constant.
(a) Show that if p has degree at most 3 then it is irreducible if and only if it has no roots
in R.
(b) Give an example of a ring R and a polynomial p(x) such that
{a ∈ R | p(a) = 0} > deg p(x).
(c) Show that if R is a field and p(x) is irreducible then (p(x)) is a prime ideal.
(d) Let ϕ : R → S be an injective ring homomorphism. Show that if ϕ(p(x)) is irreducible
in S[x] then p(x) is irreducible in R[x]. Give a counterexample to this if ϕ is not
injective.
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