# Which notation is best for $R/I$

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?

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

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.
• 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. May 5, 2020 at 1:26