In a commutative ring $R$ with identity, if $r, s \in R$ such that $(r) = (s)$, I am concluding that $r= s$ as $(r) = (s)$ implies that $s$ is an associate of $r$ and thus $r= s$. I'm not $100\%$ sure if this is true but would appreciate any feedback. Thanks in advance!
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2(Assuming $(r)$ represents the ideal generated by $r$): Not this is not true. Consider that $r$ and $-r$ both generate the same ideal, but are not equal. In general, two nonequal elements can be associates. – xxxxxxxxx Nov 19 '19 at 06:33
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1Why do you think being associate implies equality? – Bill Dubuque Nov 19 '19 at 06:39