MATH 402: ADVANCED ALGEBRA II

HOMEWORK ASSIGNMENT 2

(1) Let R be a commutative ring with identity. Let f =∑n

i=0 aixi ∈ R[x]. Show that f

is a unit in R[x] if and only if a0 ∈ R∗ and a1, · · · , an are nilpotent elements in R.Here R∗ denotes the group of units in R.

(2) Let R be a commutative ring and let ∅ 6= S ⊂ R be a multiplicative subset. LetT be a commutative ring with identity. If f : R → T is any ring homomorphismsuch that f(s) is a unit in T for all s ∈ S then show that there exists a unique ringhomomorphism f̄ : S−1R → T such that f̄φS = f . Here φS is the canonical ringhomomorphism R → S−1R defined in class. You must show the existence and theuniqueness of f̄ .This is the universal property satisfied by the localization of R with respect to S.

(3) Let R be a commutative ring with identity and let ∅ 6= S ⊂ R∗ be a multiplica-tive subset. Show that the canonical ring homomorphism φS : R → S−1R is anisomorphism. [Hint: you may try to use the universal property from Exercise 2.]

(4) Recall that a commutative ring with identity is called local if it has a unique maximalideal. Prove that R is local if and only if R \R∗ is an ideal in R.

(5) Let R be a commutative ring and let x ∈ R. Consider the multiplicative subsetS = {xn}n>0. The ring S−1R is called the localization of R with respect to x anddenoted by Rx. If R = Z/36Z then find R3.

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