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What does the notation (a) = (b) mean? For context, a and b are elements of a ring. I have tried to find an answer by googling, but it's difficult to do this since I am not sure what to call it.

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  • $\begingroup$ These two elements of the ring generate the same principle ideal. $\endgroup$ – vertical.void Jan 24 '14 at 4:48
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In a ring $\,R\,$ the notation $(a)$ denotes $aR,\,$ the (principal) ideal generated by $\,a.\,$ Thus $(a) = (b)$ means that $aR = bR,\,$ or, equivalently, $\,a\mid b\,$ and $\,b\mid a,\,$ i.e. $\,a\,$ and $\,b\,$ are associates. If, furthermore, $\,a\,$ and $\,b\,$ are cancellable, then this is equivalent to $\,a,b\,$ being strong associates, i.e. $\,a = bu\,$ for some unit $\,u\,$ (but this may fail without cancellability).

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In Algebra,Hungerford which is a standard book in ring theory, (a) is the principal ideal generated by a

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    $\begingroup$ This notation arises in other texts like Artin and Dummit & Foote as well. $\endgroup$ – Lost Jan 24 '14 at 4:38

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