Krystal Villanueva

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?

Let A be a ring, S a multiplicatively closed subset. Is it true that

uporabah5pn

Beginner2022-02-16Added 11 answers

Set $A={\mathbb{Z}}_{6}\text{}\text{and}\text{}S=A3{\mathbb{Z}}_{6}$ . 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 ${S}^{-1}A$ .)