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# 数学代写|数论代写Number Theory代考|Submodules and quotient modules

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## 数学代写|数论代写Number Theory代考|Submodules and quotient modules

Again, throughout this section, $R$ denotes a ring. The notions of subgroups and quotient groups extend in the obvious way to $R$-modules.

Definition 14.3. Let $M$ be an $R$-module. A subset $N$ is a submodule of $M$ if
(i) $N$ is a subgroup of the additive group $M$, and
(ii) $N$ is closed under scalar multiplication; that is, for all $a \in R$ and $\alpha \in N$, we have $a \alpha \in N$.

It is easy to see that a submodule $N$ of an $R$-module $M$ is also an $R$ module in its own right, with addition and scalar multiplication operations inherited from $M$.

Expanding the above definition, we see that a subset $N$ of $M$ is a submodule if and only if for all $a \in R$ and all $\alpha, \beta \in N$, we have
$$\alpha+\beta \in N, \quad-\alpha \in N, \text { and } a \alpha \in N$$
Observe that the condition $-\alpha \in N$ is redundant, as it is implied by the condition $a \alpha \in N$ with $a=-1_R$.

For $m \in \mathbb{Z}$, it is easy to see (verify) that not only are $m M$ and $M{m}$ subgroups of $M$ (see Theorems 8.6 and 8.7), they are also submodules of $M$. Moreover, for $a \in R, a M:={a \alpha: \alpha \in M}$ and $M{a}:={\alpha \in M: a \alpha=$ $\left.0_M\right}$ are also submodules of $M$ (verify).

## 数学代写|数论代写Number Theory代考|Module homomorphisms and isomorphisms

Again, throughout this section, $R$ is a ring. The notion of a group homomorphism extends in the obvious way to $R$-modules.

Definition 14.4. Let $M$ and $M^{\prime}$ be modules over $R$. An R-module homomorphism from $M$ to $M^{\prime}$ is a map $\rho: M \rightarrow M^{\prime}$, such that
(i) $\rho$ is a group homomorphism from $M$ to $M^{\prime}$, and
(ii) for all $a \in R$ and $\alpha \in M$, we have $\rho(a \alpha)=a \rho(\alpha)$.
An $R$-module homomorphism is also called an $R$-linear map. We shall use this terminology from now on. Expanding the definition, we see that a map $\rho: M \rightarrow M^{\prime}$ is an $R$-linear map if and only if $\rho(\alpha+\beta)=\rho(\alpha)+\rho(\beta)$ and $\rho(a \alpha)=a \rho(\alpha)$ for all $\alpha, \beta \in M$ and all $a \in R$.

Since an $R$-module homomorphism is also a group homomorphism on the underlying additive groups, all of the statements in Theorem 8.20 apply. In particular, an $R$-linear map is injective if and only if the kernel is trivial (i.e., contains only the zero element). However, in the case of $R$-module homomorphisms, we can extend Theorem 8.20, as follows:
Theorem 14.5. Let $\rho: M \rightarrow M^{\prime}$ be an $R$-linear map.
(i) For any submodule $N$ of $M, \rho(N)$ is a submodule of $M^{\prime}$.
(ii) $\operatorname{ker}(\rho)$ is a submodule of $M$.
(iii) For any submodule $N^{\prime}$ of $M^{\prime}, \rho^{-1}\left(N^{\prime}\right)$ is a submodule of $M$.

## 数学代写|数论代写Number Theory代考|Submodules and quotient modules

(一) $N$ 是加法群的一个子群 $M$, 和
(ii) $N$ 在标量乘法下是封闭的； 也就是说，对于所有 $a \in R$ 和 $\alpha \in N$ ，我们有 $a \alpha \in N$.

$$\alpha+\beta \in N, \quad-\alpha \in N, \text { and } a \alpha \in N$$

## 数学代写|数论代写Number Theory代考|Module homomorphisms and isomorphisms

(i) $\rho$ 是群同态从 $M$ 到 $M^{\prime} ，$ 并且
(ii) 对于所有 $a \in R$ 和 $\alpha \in M$ ，我们有 $\rho(a \alpha)=a \rho(\alpha)$.

(i) 对于任何子模块 $N$ 的 $M, \rho(N)$ 是一个子模块 $M^{\prime}$.
(二) $\operatorname{ker}(\rho)$ 是一个子模块 $M$.
(iii) 对于任何子模块 $N^{\prime}$ 的 $M^{\prime}, \rho^{-1}\left(N^{\prime}\right)$ 是一个子模块 $M$.

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