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# 数学代写|抽象代数代写Abstract Algebra代考|Basic properties of rings

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## 数学代写|抽象代数代写Abstract Algebra代考|Basic properties of rings

Believe it or not, even with all we’ve accomplished, we still haven’t developed a theory to solve equations as simple as $\frac{1}{2} x+1=0$. That’s because the expression $\frac{1}{2} x+1$ isn’t just about multiplication or addition: it involves both operations. Group theory is all about the properties of sets with a single binary operation, so group theory won’t provide the means to solve this type of linear equation. That means we need to develop a new algebraic structure that comes with more than one operation.

Definition 12.1. Let $R$ be a set with two binary operations on $R$, called addition and denoted + , and multiplication and denoted $\cdot$ Then $\langle R,+, \cdot\rangle$ is a ring if and only if the following hold:
(1) $\langle R,+\rangle$ is an abelian group.
(2) $\langle R, \cdot\rangle$ is an associative binary structure.
(3) $a \cdot(b+c)=a \cdot b+a \cdot c$ and $(b+c) \cdot a=b \cdot a+c \cdot a$ for all $a, b, c \in R$ (the distributive laws).

Before we proceed with examples, some observations about this definition are in order. First, we do just write $a b$ to mean $a \cdot b$ as we did with groups under multiplication, and we simply say that $R$ is a ring without explicitly mentioning the two operations. Second, notice that multiplication in a ring is very, very unstructured. Multiplication is only closed and associative: there need not be a multiplicative identity, and even if there is, there need not be any multiplicative inverses, and just like groups, multiplication need not be commutative. In particular, that means that we should not expect to be able to solve equations involving multiplication, since we need inverses to be able to “undo” an operation. Third, a ring’s additive structure is really, really nice: an abelian group! Addition commutes, and in fact we can now talk about subtraction in a ring by defining $a-b=a+(-b)$. Because of that, we also introduce a notation for the additive identity element: we use the symbol $\mathbf{0}$ to denote the additive identity.

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

Just as we did with groups, we need to develop a notion of maps between rings that preserve structure. Now that we have two operations to deal with, those maps need to preserve both structures.

Definition 12.18. Let $R$ and $R^{\prime}$ be rings and $\phi: R \rightarrow R^{\prime}$ be a function. Then $\phi$ is a (ring) homomorphism if $\phi(a+b)=\phi(a)+\phi(b)$ and $\phi(a b)=\phi(a) \phi(b)$ for all $a, b \in R$. A ring homomorphism $\phi$ is an isomorphism if and only if $\phi$ is also a bijection, and a ring automorphism is a ring isomorphism from a ring to itself. Two rings are isomorphic if and only if there is a ring isomorphism from one ring to the other.

Notice that since ring homomorphisms and isomorphisms are, in fact, group homomorphisms and group isomorphisms under addition, all of the results from group theory still apply to the additive structure of a ring. On the other hand, since the multiplicative structure of a ring isn’t a group structure, we don’t get all of the nice multiplicative properties we might hope for. However, if you have a ring isomorphism, then all of the algebraic properties like commutativity, unity, and inverses are preserved, as well as unity and inverses being mapped to unity and inverses. Yet even with ring homomorphisms, we do get a few nice properties similar to our results from groups.
Theorem 12.19. Let $\phi: R \rightarrow R^{\prime}$ be a ring homomorphism.
(1) $\phi\left(a^n\right)=\phi(a)^n$ for all $a \in R$ and $n \in \mathbb{Z}^{+}$.
(2) If $S$ is a subring of $R$, then $\phi(S)$ is a subring of $R^{\prime}$, and if $S$ is commutative, then $\phi(S)$ is also commutative.
(3) If $S^{\prime}$ is a subring of $R^{\prime}$, then $\phi^{-1}\left(S^{\prime}\right)$ is a subring of $R$.

## 数学代写|抽象代数代写Abstract Algebra代考|Basic properties of rings

12.1.定义设$R$是一个集合，在$R$上有两个二进制运算，分别是加法运算，记为+，和乘法运算，记为$\cdot$，那么$\langle R,+, \cdot\rangle$是一个环，当且仅当以下条件成立:
(1) $\langle R,+\rangle$是一个阿贝尔群。
(2) $\langle R, \cdot\rangle$是一个结合二元结构。
(3)所有$a, b, c \in R$(分配律)为$a \cdot(b+c)=a \cdot b+a \cdot c$和$(b+c) \cdot a=b \cdot a+c \cdot a$。

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

12.18.定义让 $R$ 和 $R^{\prime}$ 他响了铃， $\phi: R \rightarrow R^{\prime}$ 是一个函数。然后 $\phi$ 环是同态的吗 $\phi(a+b)=\phi(a)+\phi(b)$ 和 $\phi(a b)=\phi(a) \phi(b)$ 对所有人 $a, b \in R$． 一个环同态 $\phi$ 同构是否当且仅当 $\phi$ 也是一个双射，一个环自同构是一个环到它自己的环同构。两个环是同构的当且仅当从一个环到另一个环存在环同构。

(1)所有$a \in R$和$n \in \mathbb{Z}^{+}$为$\phi\left(a^n\right)=\phi(a)^n$。
(2)如果$S$是$R$的子带，则$\phi(S)$是$R^{\prime}$的子带;如果$S$是可交换的，则$\phi(S)$也是可交换的。
(3)如果$S^{\prime}$是$R^{\prime}$的子带，那么$\phi^{-1}\left(S^{\prime}\right)$就是$R$的子带。

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