Which notation is best for $R/I$

Question

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?

Answer 1

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.