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# 数学代写|抽象代数代写Abstract Algebra代考|MATH3303 Uniqueness of the Unity and Inverses

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## 数学代写|抽象代数代写Abstract Algebra代考|Uniqueness of the Unity and Inverses

If a ring has a unity, it is unique. If a ring element has a multiplicative inverse, it is unique.
Many students have the mistaken tendency to treat a ring as if it were a group under multiplication. It is not. The two most common errors are the assumptions that ring elements have multiplicative inverses – they need not-and that a ring has a multiplicative identity-it need not. For example, if $a, b$, and $c$ belong to a ring, $a \neq 0$ and $a b=a c$, we cannot conclude that $b=c$.Similarly, if $a^{2}=a$, we cannot conclude that $a=0$ or 1 (as is the case with real numbers). In the first place, the ring need not have multiplicative cancellation, and in the second place, the ring need not have a multiplicative identity. There is an important class of rings that contains $Z$ and $Z x]$ wherein multiplicative identities exist and for which multiplicative cancellation holds. This class is taken up in the next chapter.

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

$A$ nonempty subset $S$ of a ring $R$ is a subring if $S$ is closed under subtraction and multiplication – that is, if $a-b$ and $a b$ are in $S$ whenever $a$ and $b$ are in $S$.
PROOF Since addition in $R$ is commutative and $S$ is closed under subtraction, we know by the One-Step Subgroup Test (Theorem 3.1) that $S$ is an Abelian group under addition. Also, since multiplication in $R$ is associative as well as distributive over addition, the same is true for multiplication in $S$. Thus, the only condition remaining to be checked is that multiplication is a binary operation on $S$. But this is exactly what closure means.

We leave it to the student to confirm that each of the following examples is a subring.

I EXAMPLE $8{0}$ and $R$ are subrings of any ring $R$. ${0}$ is called the trivial subring of $R$.

IEXAMPLE $9{0,2,4}$ is a subring of the ring $Z_{6}$, the integers modulo 6. Note that although 1 is the unity in $Z_{6}, 4$ is the unity in { $0,2,4}$.
I EXAMPLE 10 For each positive integer $n$, the set
$$n Z={0, \pm n, \pm 2 n, \pm 3 n, \ldots}$$

## 数学代写|由象代数代写Abstract Algebra代考|Subring Test

$A$ 非空子集 $S$ 个个戒指 $R$ 如果是子环 $S$ 在减㹤和乘法下是闭合的一一也就是说，如果 $a-b$ 和 $a b$ 在 $S$ 每当 $a$ 和 $b$ 在 $S$.

| 例 10 对于每个正整数 $n$, 集合
$$n Z=0, \pm n, \pm 2 n, \pm 3 n, \ldots$$

## MATLAB代写

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