## Description

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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