# Clarification of Notation: Ideals of a quotient ring

I am working through Abstract Algebra by Dummit and Foote and want to make sure I understand the notation used in the text. The text is discussing the Isomorphism Theorems for Rings and uses the following example.

"Let $$R= \mathbb{Z}$$ and let $$I$$ be the ideal $$12\mathbb{Z}$$. The quotient ring $$\bar{R} = R/I= \mathbb{Z}/12\mathbb{Z}$$ has ideals $$\bar{R}, 2\mathbb{Z}/12\mathbb{Z}, 3\mathbb{Z}/12\mathbb{Z}, 4\mathbb{Z}/12\mathbb{Z}, 6\mathbb{Z}/12\mathbb{Z}$$ and $$\bar{0} = 12\mathbb{Z}/12\mathbb{Z}$$ corresponding to the ideals in $$R = \mathbb{Z}, 2\mathbb{Z}, 3\mathbb{Z}, 4\mathbb{Z}, 6\mathbb{Z}$$ and $$12\mathbb{Z} = I$$ of $$R$$ containing $$I$$, respectively. "

What is the meaning of $$2\mathbb{Z}/12\mathbb{Z}$$? More generally what does $$d\mathbb{Z}/n\mathbb{Z}$$ mean when $$d$$ is a divisor of $$n$$? I know that $$\mathbb{Z}/12\mathbb{Z}$$ is the set of integers modulo 12. Would $$2\mathbb{Z}/12\mathbb{Z}$$ then be the multiples of 2 in the set of modulo 2? In other words would $$2\mathbb{Z}/12\mathbb{Z} = \{\bar{2}, \bar{4}, \bar{6}, \bar{8}, \bar{10}\}$$.

While @mathslearner98 gives a very nice and very concrete answer, I thought I could add a slightly more high-level answer, to give more context…

Consider a ring $$R$$ and an ideal $$I$$ in $$R$$. This gives rise to a quotient map $$\pi:R \rightarrow R/I$$.

Given any subset $$S\subseteq R$$ the notation $$S/I$$ is a very convenient notation for the image $$\pi(S)\subseteq S$$ of $$S$$ under $$\pi$$, since it reminds us that such an image is of the form $$\pi(S)=S/I=\{s+ I \mid s\in S\}.$$

Now since $$\pi$$ is surjective we find that every ideal $$J\subseteq R$$ maps to an ideal $$\pi(J)=J/I\subseteq S$$. This is precisely what the $$d\Bbb Z/n\Bbb Z$$ is expressing.

The reason for why we assume that $$d$$ divides $$n$$ is the following. Ideals in $$R/I$$ are in bijection to those ideals in $$R$$, which contain $$I$$. In other words the maps $$\begin{array}{rcl} \{\text{ideals }J \subseteq R \mid I \subseteq J\}&\longrightarrow& \{\text{ideals } \widetilde{J}\subseteq R/I\}\\ J & \mapsto & \pi(J)=J/I\\ \pi^{-1}(\widetilde{J}) & \leftarrow & \widetilde{J} \end{array}$$ are mutually inverse. So in our case we not only know that every ideal in $$\Bbb Z$$ gives an ideal in $$\Bbb Z/n\Bbb Z$$, but that every ideal in $$\Bbb Z/n\Bbb Z$$ is of the form $$d\Bbb Z/ n\Bbb Z$$ for some ideal $$d\Bbb Z$$ in $$\Bbb Z$$ containing $$n\Bbb Z$$, hence for some $$d$$ dividing $$n$$!

$$2\mathbb{Z}/12\mathbb{Z}=\{\overline{0},\overline{2},\overline{4},\overline{6},\overline{8},\overline{10}\}$$ as a set. To get an idea to understand this sort of a notation in general, note that $$2\mathbb{Z}=\{2n:n\in\mathbb{Z}\}$$ and $$12\mathbb{Z}=\{12n:n\in\mathbb{Z}\}$$. So $$2\mathbb{Z}/12\mathbb{Z}=\{x+12\mathbb{Z}:x\in2\mathbb{Z}\}$$, and since, for instance, $$2+12\mathbb{Z}=14+12\mathbb{Z}$$, we didn't have to write $$\overline{14}$$ when writing $$2\mathbb{Z}/12\mathbb{Z}$$ as a set because $$\overline{2}$$ is already there.

So in short, a general strategy to understand $$d\mathbb{Z}/n\mathbb{Z}$$ is first to write down $$d\mathbb{Z}$$ and $$n\mathbb{Z}$$ as sets, then think about how would $$\{x+n\mathbb{Z}:x\in d\mathbb{Z}\}$$ look like.

Here, $$x+n\mathbb{Z}$$ is the set $$\{x+y:y\in n\mathbb{Z}\}$$.

• Another interpretation is $2\mathbb{Z}/12\mathbb{Z}=2(\mathbb{Z}/12\mathbb{Z}) = \{ 2x : x \in 12\mathbb{Z} \}$.
– lhf
Nov 5, 2021 at 10:40