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Math 512 Problem Set 1 

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Math 512 Problem Set 1
Exercise 1. Let R be an integral domain, and let T be an integral domain
such that R ⊂ T ⊂ Frac R. Show that Frac R = Frac T.
Exercise 2. Let R be an integral domain, and S ⊂ R a multiplicative
subset that does not contain 0. Show that if R is a PID, then so is S
−1R.
Exercise 3. Let R be a commutative unital ring, S ⊂ R a multiplicative
subset, and I ⊂ R an ideal. Show that S
−1

I =

S−1

I (Recall that
I = {x ∈ R | x
n ∈ I for some n ∈ N}).
Exercise 4. Let R be a commutative unital ring. Show that R is local if
and only if whenever r + s = 1, then either r ∈ R∗ or s ∈ R∗
Exercise 5. Show that every nonzero homomorphic image of a local ring
is local.
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