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I am in a abstract algebra course and I am not sure what terms like $\mathbb{Z}_{6}$, $\mathbb{Z}_{5}$, $\mathbb{Z}_{n}$, etc, mean. Could someone help explain to me what they mean and stand for? Thanks!

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3 Answers 3

If you've reached rings and ideals, you know that $n\mathbb{Z}$ is an ideal of $\mathbb{Z}$. The quotient ring $\mathbb{Z}/n\mathbb{Z}$ is sometimes denoted $\mathbb{Z}_n$.

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That is standard notation about quotient groups, or rings, although if it's your first course in asbtract algebra you're probably only analising them as groups.

$$\mathbb{Z}_n=\mathbb{Z}/n\mathbb{Z}$$

As you should know, $gG$ stands for the set of all elements og the group multiplied by the element $g$, and so $n\mathbb{Z}$ is the multiples of $n$. The quotient group is formed by all equivalence classes formed by:

$$x\sim y\;\;\text{iff}\;\;x-y\in n\mathbb{Z}$$

So every class is formed by a number from $0$ to $n-1$ and all the multiples of $n$ added to that number:

$$[0]=\left\lbrace 0,n,-n,2n,-2n,3n,-3n,...\right\rbrace$$ $$[1]=\left\lbrace 1,n+1,-n+1,2n+1,-2n+1,3n+1,-3n+1,...\right\rbrace$$

And so on. Usually this classes arenamed by the number: $[0]\equiv 0$, and they're used like numbers in which addition is cyclic, it goes back to 0 after $n-1$, and so:

$$\mathbb{Z}_n=\left\lbrace 0,1,2,...,n-2,n-1\right\rbrace$$

I've seen some books call this: In module $n$ there exist only $n$ integer numbers.

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Stricly speaking, $\mathbb{Z}_n$ stands for $\mathbb{Z}/n\mathbb{Z}$ which consists of the sets of numbers that leave the same remainder when divided by $n$. For example $\mathbb{Z}_2$ consists of two things: all of the odd numbers and all of the even number. With $\mathbb{Z}_3$ we have three things: the multiples of three, the numbers one more that a multiple of three and the numbers two more than a multiple of three.

In practice, you can think of $\mathbb{Z}_n$ as the set $\{0,1,2,\ldots,n-1\}$, where the numbers represent the remainders. Additions and multiplication are defined by always reducing the answer to its remainder when divided by $n$. For example, we can think of $\mathbb{Z}_6$ as $\{0,1,2,3,4,5\}$. Then $3 \times 3 = 9 \equiv 3$ because $9$ leaves remainder $3$ when divided by $6$. Similarly, $5+5 = 10 \equiv 4$ because $10$ leaves remainder $4$ when divided by $6$.

This arithmetic is called modular arithmetic. When we reduce a number to its remainder when divided by, for example $6$, we write things like $9 \equiv 3 \bmod 6$ and $10 \equiv 4 \bmod 6$.

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