"Very basic question about localizations Let A be..." solved and explained - Plainmath

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Algebra
Abstract algebra

Krystal Villanueva

Answered question

2022-02-15

Very basic question about localizations
Let A be a ring, S a multiplicatively closed subset. Is it true that

\frac{a}{1} \in S^{- 1} A

is a non zero-divisor if and only if

a \in A

is?

Answer & Explanation

uporabah5pn

Beginner2022-02-16Added 11 answers

Set

A = Z_{6} and S = A 3 Z_{6}

. Then

\hat{2}

is a zerodivisor in A and

\frac{\hat{2}}{\hat{1}}

is not zerodivisor in

S^{- 1} A

. (Actually it is invertible in

S^{- 1} A

.)

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