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When $R$ is a ring and $I$ is an ideal of $R$, I have seen a variety of notational uses for the cosets in $R/I$, and I'm not sure which one is best in which context. For $a\in R$, if $C_a\in R/I$ is the coset of $I$ containing $a$, then here is a list of a few I've seen: \begin{align} C_a&=a+I\\ C_a&=a\bmod I\\ C_a&=\bar a \end{align} Additionally, in my own personal use, when I need to distinguish between elements of different quotient rings, I often use the notation $C_a=(a)_I$. Is there any others in common use?

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    $\begingroup$ The left coset is $aI$ and the right coset is $Ia$ for $a\in R$. If we want the additive notion, then $a+I$. $\endgroup$ Apr 29, 2020 at 20:25
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    $\begingroup$ Agree with @DietrichBurde (+1 for him). Also the notation $[a]$ isn't uncommon. $\endgroup$
    – MPW
    Apr 29, 2020 at 20:35
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    $\begingroup$ @Dietrich_Burde What about converting your comment into an answer, possibly including MPW remark about $[a]$? $\endgroup$
    – J.-E. Pin
    Apr 30, 2020 at 5:11

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All three notations are fine. Here’s my take on when to use which:

  1. Use $a + I$ when defining $R/I$ and when proving the basic properties. Also use it if you want to emphasize which ideal $I$ is relevant (for example, because several ideals are in use).

  2. Use $a \bmod I$ if the appearance of cosets might be a “surprise” to the reader or if the ideal $I$ is a complicated expression itself (which might make the notation $a + I$ confusing).

  3. Use $\overline{a}$ if you are performing computations in the ring $R/I$ and the reasons in 1. don’t apply, i.e. if there is no possible confusing about the ideal $I$. In this case, this notation is shortest and usually makes the underlying algebra easiest to read.

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  • $\begingroup$ Makes sense. I didn't think about if $I$ was defined in a complicated way, or if cosets might be a surprising turn of reasoning. Thanks for this. $\endgroup$
    – JasonM
    May 5, 2020 at 1:26

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