Confusion about notation

Question

Is $Z_{11}$ same as $\frac{Z}{11Z}$. I have been said in class that latter contains equivalence classes, but in assignment i have seen $Z_{11}$. Can someone clarify difference between two ?

Thanks

Answer 1

Both of the notations $\mathbb Z_{11}$ and $\mathbb Z/11 \mathbb Z$ are commonly used for the ring of integers modulo $11$.

The notation $\frac{\mathbb Z}{11\mathbb Z}$ is less common; quotients of algebraic structures are almost always written with a slash, not with a fraction bar.

With the notation $\mathbb Z/11\mathbb Z$ the elements of the ring are invariably constructed as equivalence classes (because that's what the quotient notation means). Authors who write $\mathbb Z_{11}$ may either have defined the ring in the same way, or ad hoc as $\{0,1,\ldots,10\}$ with special arithmetic operations, as Yoyostein describes.

Other notations for this include $\mathbb Z/(11)$, $\mathbb Z/\langle 11 \rangle$, or $\mathbb F_{11}$ if you want to emphasize that it's a field (every nonzero element has an inverse modulo $11$).

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There's a different possible meaning of $\mathbb Z_{11}$, namely the ring of 11-adic integers. Some people prefer the longer notation for the ring of integers modulo $11$ in order to avoid confusion with this.

Answer 2

Technically, they are different as sets.

$\mathbb{Z}_{11}=\{0, 1, 2, \dots, 9, 10\}$

Each element is a number. Addition is done modulo 11.

$\mathbb{Z}/11\mathbb{Z}=\{0+11\mathbb{Z}, 1+11\mathbb{Z}, \dots, 10+11\mathbb{Z}\}$

Each element is a coset (which is an equivalence class like what you said).

However, as groups they are isomorphic.

Other notations you may see are $\mathbb{Z}/11$, or $C_{11}$.