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# 数学代写|抽象代数代写Abstract Algebra代考|MATH412 Integral domains

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## 数学代写|抽象代数代写Abstract Algebra代考|Integral domains

Example 27.1. In $\mathbb{Z}{12}$, there is a pair of non-zero elements whose product is zero: $3 \cdot 4=$ 0 , for instance. Recall that such elements are called zero divisors in $\mathbb{Z}{12}$. Other zero divisors in $\mathbb{Z}{12}$ include $2,6,8,9,10$. Note that $2 \cdot 6=0,8 \cdot 9=0$, and $6 \cdot 10=0$ in $\mathbb{Z}{12}$.
Some rings do not have any zero divisors, which motivates the following definition.
Definition 27.2 (Integral domain). A commutative ring is called an integral domain if it does not contain any zero divisors.

Example 27.3. If $a$ and $b$ are non-zero integers, then their product $a b$ is also non-zero. Hence, there are no zero divisors in $\mathbb{Z}$, which implies that $\mathbb{Z}$ is an integral domain.
Example 27.4. In the ring $\mathbb{Z}{13}={0,1,2, \ldots, 12}$, every non-zero element is a unit, i.e., an element with a multiplicative inverse. Then by Theorem $26.18$, these units cannot be zero divisors. Thus, there are no zero divisors in $\mathbb{Z}{13}$, so that $\mathbb{Z}_{13}$ is an integral domain. The same argument shows that the set of real numbers $\mathbb{R}$ is an integral domain as well.

## 数学代写|抽象代数代写Abstract Algebra代考|Fields

Example 27.12. In $\mathbb{Z}_7={0,1,2,3,4,5,6}$, all non-zero elements are units; i.e., they have multiplicative inverses. Note that $1 \cdot 1=1,2 \cdot 4=1,3 \cdot 5=1$, and $6 \cdot 6=1$ in $\mathbb{Z}_7$. Similarly, all non-zero elements of $\mathbb{R}$ are units. While a ring is never a multiplicative group, since the additive identity element 0 does not have a multiplicative inverse, examples like $\mathbb{Z}_7$ and $\mathbb{R}$ come awfully close!

Rings like $\mathbb{Z}_7$ and $\mathbb{R}$, which are almost multiplicative groups, are examples of a field.

Definition $27.13$ (Field). A commutative ring is called a field if every non-zero element is a unit.

Examples of a field include $\mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}7$, and $\mathbb{Z}{13}$. Below are a couple more examples.

## MATLAB代写

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