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Proposition: Criterions for Equality of Principal Ideals
Two principal ideals $(a)$ and $(b)$ of an integral domain $(R, + ,\cdot)$ are equal, if and only if $a$ and $b$ are associates in $R,$ formally $$(a)=(b)\Longleftrightarrow a\sim b.$$
A principal ideal $(a)$ equals the zero ring, if and only if $a=0,$ formally $$(a)=(0)\Longleftrightarrow a=0.$$
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References
Bibliography
 Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013