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Some definition like:

"Let $I(R)$ be the set of all possible ideals of the ring $R$"

would be very useful, e.g. for saying "$(2) \in I(\mathbb{Z})$" instead of "$(2)$ is an ideal of $\mathbb{Z}$".

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    $\begingroup$ I'm not sure whether there is any standard notation for the set of all ideals of a ring. You can define it in the context that you are working on. The only thing is that you have to say it in the begining that you are going to use this notation for the set of all ideals. On the other hand, there are some standard notations for prime ideals and maximal ideals. For a commutative ring $A$ with unity, the set of all prime ideals is denoted by $\text{Spec}(A),$ called the spectrum of $A$ or the prime spectrum of $A$ and for the set of maximal ideals $\text{max}(A), \text{m-Spec}(A).$ $\endgroup$ – Krish Mar 15 '15 at 18:50
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There does not seem to be a standard notation, but something I-ish sounds reasonable. This is what a quick search gave me:

  • $\mathcal I_R$, $\mathcal I(R)$ $\qquad$ R.Y. Sharp, Steps in Commutative Algebra, starting from remark 2.39; some other papers and lecture notes.
  • $\mathcal{ID}(F)$ $\qquad$ Hertling, Hoffman et al., Reliable Implementation of Real Number Algorithms: Theory and Practice, from above Lemma 9, p. 209
  • $\operatorname{Id} L$ $\qquad$ (more generally for lattices), Gierz, Hofmann et al., A Primer on Complete Lattices, definition 1.3

One also encounters $I$ for the set of fractional ideals, so this notation is not free of ambiguity.

Basically, it seems that "the set of all ideals" is not used often enough to have its widespread standard notation, unlike $\operatorname{Spec}$ and $\operatorname{Max(Spec)}$. That said, you are free to introduce new notations in your work, and there's no harm in repeating every now and then what it stands for, as in "we conclude that the set of all ideals $I(R)$ satisfies ..."

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  • $\begingroup$ Bonus points for dedication to answering a three years old question. $\endgroup$ – Jorge Rodriguez Mar 12 '18 at 21:00
  • $\begingroup$ I made it CW, so any rep goes to my karma... $\endgroup$ – Bart Michels Mar 13 '18 at 21:15

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