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# 数学代写|交换代数代写Commutative Algebra代考|MATH483 Definition of a Module

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## 数学代写|交换代数代写Commutative Algebra代考|Definition of a Module

Definition (3.1.1). – Let A be a ring. A right A-module is a set $\mathrm{M}$ endowed with two binary laws: an internal addition, $\mathbf{M} \times \mathrm{M} \rightarrow \mathbf{M},(m, n) \mapsto m+n$, and a scalar multiplication $\mathrm{M} \times \mathrm{A} \rightarrow \mathbf{M},(m, a) \mapsto m a$, subject to the following properties:
(i) Axioms concerning the addition law and stating that $(\mathrm{M},+)$ is an abelian group:

For every $m, n$ in $\mathbf{M}, m+n=n+m$ (commutativity of addition);

For every $m, n, p$ in $\mathbf{M},(m+n)+p=m+(n+p)$ (associativity of addition);

There exists an element $0 \in \mathrm{M}$ such that $m+0=0+m=m$ for every $m \in \mathrm{M}$ (neutral element for the addition);

For every $m \in \mathbf{M}$, there exists an $n \in \mathbf{M}$ such that $m+n=n+m=0$ (existence of an additive inverse);
(ii) Axioms concerning the multiplication law:

For any $m \in \mathrm{M}, m 1=m$ (the unit of $\mathrm{A}$ is neutral for the scalar multiplication);

For any $a, b \in \mathrm{A}$ and any $m \in \mathrm{M}$, one has $m(a b)=(m a) b$ (associativity of the scalar multiplication);
(iii) Axioms relating the ring addition, the module addition and the scalar multiplication:

For any $a, b \in \mathrm{A}$ and any $m \in \mathrm{M}$, one has $m(a+b)=m a+m b$ (scalar multiplication distributes over the ring addition);

For any $a \in \mathrm{A}$ and any $m, n \in \mathrm{M}$, one has $(m+n) a=m a+n a$ (scalar multiplication distributes over the module addition).

Analogously, a left A-module is a set $\mathrm{M}$ endowed with two binary laws: an internal addition, $\mathrm{M} \times \mathrm{M} \rightarrow \mathrm{M},(m, n) \mapsto m+n$, and a scalar multiplication $\mathrm{A} \times \mathrm{M} \rightarrow \mathrm{M},(a, m) \mapsto a m$, subject to the following properties:
(i) Axioms concerning the addition law and stating that $(\mathrm{M},+)$ is an abelian group:

for every $m, n$ in M, $m+n=n+m$ (commutativity of addition);

for every $m, n, p$ in M, $(m+n)+p=m+(n+p)$ (associativity of addition);

there exists an element $0 \in \mathrm{M}$ such that $m+0=0+m=m$ for every $m \in M$ (neutral element for the addition);

for every $m \in \mathbf{M}$, there exists $n \in \mathbf{M}$ such that $m+n=n+m=0$ (existence of an additive inverse);

## 数学代写|交换代数代写Commutative Algebra代考|Let A be a ring and let M be an A-module

Let $m \in$ M. The set $\operatorname{Ann}{\mathrm{A}}(m)$ of all $a \in$ A such that $m a=0$ is a right ideal of $\mathrm{A}$, called the annihilator of $m$. (Indeed, for $a, b \in \mathrm{A}$, one has $m 0=0$, $m(a+b)=m a+m b=0$ if $m a=m b=0$, and $m(a b)=(m a) b=0$ if $m a=0$.) The set $A n n{\mathrm{A}}(\mathrm{M})$ of all $a \in \mathrm{A}$ such that $m a=0$ for all $m \in \mathrm{M}$ is called the annihilator of $\mathrm{M}$. Since it is the intersection of all $\operatorname{Ann}{\mathrm{A}}(m)$, for $m \in \mathbf{M}$, it is a right ideal of $\mathrm{A}$. On the other hand, if $a \in \operatorname{Ann}{\mathrm{A}}(\mathrm{M})$ and $b \in \mathrm{A}$, one has $m(b a)=(m b) a=0$ for all $m \in \mathbf{M}$, so that $b a \in \operatorname{Ann}{\mathrm{A}}(\mathrm{M})$. Consequently, $\operatorname{Ann}{\mathrm{A}}(\mathrm{M})$ is a two-sided ideal of $\mathrm{A}$.
If $\operatorname{Ann}{\mathrm{A}}(\mathrm{M})=0$, one says that the A-module $\mathrm{M}$ is faithful. In fact, the ideal $\operatorname{Ann}{\mathrm{A}}(\mathrm{M})$ is the kernel of the morphism $\mu: \mathrm{A} \rightarrow \operatorname{End}(\mathrm{M})$, $a \mapsto \mu_a$. In this way, $\mathrm{M}$ can be viewed as an A/I-module, for every ideal I of A such that $I \subset \operatorname{Ann}A(M)$. Taking $I=\operatorname{Ann}_A(M)$, we observe that $M$ is a faithful $\left(\mathrm{A} / \operatorname{Ann}{\mathrm{A}}(\mathrm{M})\right)$-module.

Conversely, if $\mathrm{M}$ is an $\left(\mathrm{A} / \mathrm{Ann}{\mathrm{A}}(\mathrm{M})\right)$-module, the composition $\mathrm{A} \rightarrow$ $\mathrm{A} / \operatorname{Ann}{\mathrm{A}}(\mathrm{M}) \rightarrow$ End(M) endows $\mathrm{M}$ with the structure of an A module whose annihilator contains $\operatorname{Ann}_{\mathrm{A}}(\mathrm{M})$.

Definition (3.1.6). – Let $\mathrm{A}$ be a ring and let $\mathrm{M}$ be a right A-module. $\mathrm{A}$ submodule of $\mathrm{M}$ is a subset $\mathrm{N}$ of $\mathrm{M}$ satisfying the following properties:
(i) $\mathrm{N}$ is an abelian subgroup of $\mathrm{M}$;
(ii) if $a \in \mathrm{A}$ and $m \in \mathrm{M}$, then $m a \in \mathrm{N}$.
There is a similar definition for left A-modules, and for (A, B)-bimodules.
Examples (3.1.7). – a) Let $\mathrm{M}$ be a A-module. The subsets ${0}$ and $\mathrm{M}$ of $\mathrm{M}$ are submodules.
$b$ ) Let $\mathrm{A}$ be a ring. The left ideals of $\mathrm{A}$ are exactly the submodules of the left A-module $\mathrm{A}_{\mathrm{s}}$. The right ideals of $\mathrm{A}$ are the submodules of the right A-module $\mathrm{A}_d$.
c) Let A be a division ring. The submodules of an A-vector space are its vector subspaces.

## 数学代写|交换代数代写Commutative Algebra代考|Definition of a Module

(i) 关于加法定律的公理并说明 $(\mathrm{M},+)$ 是阿贝尔群:

(ii) 关于乘法定律的公理:

(iii) 与环加法、模加法和标量乘法相关的公理:

(i) 关于加法定律的公理并说明 $(\mathrm{M},+)$ 是阿贝尔群:

## 数学代写|交换代数代写Commutative Algebra代考|Let A be a ring and let M be an A-module

(i) $\mathrm{N}$ 是一个阿贝尔子群M;
(ii) 如果 $a \in \mathrm{A}$ 和 $m \in \mathrm{M}$ ， 然后 $m a \in \mathrm{N}$.

c) 设 $A$ 为分割环。 $A$ 向量空间的子模块是它的向量子空间。

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