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