Assignment 10 – Due Friday, November 21
Turn this in at the start of recitation on Friday, November 21.
1. Suppose that R is a commutative ring. Show that it satisfies a cancellation property (i.e., that ab = ac
implies b = c) if and only if it is an integral domain.
2. Suppose that R is unital and S is an integral domain and let φ : R → S be a homomorphism. Show
that either φ(1) = 0 or φ(1) = 1. On the other hand, if S is not an integral domain, then this might
not be true; find a homomorphism φ : Z6 → Z15 such that φ(1) 6= 0, 1.
3. Prove that the ring of real numbers R is not isomorphic to the ring of complex numbers C. (Amazingly,
the abelian group R is isomorphic to the abelian group C, so you can’t just say “R is one-dimensional,
but C is two-dimensional.”)
4. Herstein, p. 130: #6, #8
5. Herstein, p. 135: #2
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