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# 数学代写|交换代数代写Commutative Algebra代考|MATH662

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## 数学代写|交换代数代写Commutative Algebra代考|Bezout Rings

A ring $\mathbf{A}$ is called a Bezout ring when every finitely generated ideal is principal. This is the same as saying that every ideal with two generators is principal.
$$\forall a, b \exists u, v, g, a_1, b_1 \quad\left(a u+b v=g, a=g a_1, b=g b_1\right) .$$
A Bezout ring is strongly discrete if and only if the divisibility relation is explicit. An integral Bezout ring is called a Bezout domain.
A local ring is a ring $\mathbf{A}$ where is satisfied the following axiom
$$\forall x, y \in \mathbf{A} \quad x+y \in \mathbf{A}^{\times} \Longrightarrow\left(x \in \mathbf{A}^{\times} \text {or } y \in \mathbf{A}^{\times}\right) .$$
This is the same as asking
$$\forall x \in \mathbf{A} \quad x \in \mathbf{A}^{\times} \text {or } 1-x \in \mathbf{A}^{\times} .$$
Note that according to this definition the trivial ring is local. Moreover, the “or” must be understood in the constructive sense: the alternative must be explicit. Most of the local rings with which we usually work in classical mathematics actually satisfy the previous definition if we look at it from a constructive point of view.

## 数学代写|交换代数代写Commutative Algebra代考|Finitely Presented Modules over Valuation Rings

A matrix $B=\left(b_{i, j}\right) \in \mathbf{A}^{m \times n}$ is said to be in Smith form if every coefficient out of the principal diagonal is null, and if for $1 \leqslant i<\inf (m, n)$, the diagonal coefficient $b_{i, i}$ divides the following $b_{i+1, i+1}$.

7.2 Proposition Let $\mathbf{A}$ be a local Bezout ring.

1. Every matrix of $\mathbf{A}^{m \times n}$ is elementarily equivalent to a matrix in Smith form.
2. Every finitely presented $\mathbf{A}$-module $M$ is isomorphic to a direct sum of modules $\mathbf{A} /\left\langle a_i\right\rangle: M \simeq \bigoplus_{i=1}^p \mathbf{A} /\left\langle a_i\right\rangle$, with in addition, for each $i<p, a_{i+1}$ divides $a_i$.
D 1. We use the Gauss pivot method by choosing for first pivot a coefficient of the matrix which divides all the others. We finish by induction.
3. Direct consequence of item 1 .
Remark This result is completed by the uniqueness theorem (Theorem 5.1) as follows.
4. In the reduced matrix in Smith form the ideals $\left\langle b_{i, i}\right\rangle$ are uniquely determined.
5. In the decomposition $\bigoplus_{i=1}^p \mathbf{A} /\left\langle a_i\right\rangle$, the ideals $\left\langle a_i\right\rangle$ are uniquely determined, except that ideals in excessive numbers can be equal to $\langle 1\rangle$ : we can delete the corresponding terms, but this only happens without fail when we have an invertibility test in the ring.

A ring $\mathbf{A}$ is called a strict Bezout ring when every vector $[u v] \in \mathbf{A}^2$ can be transformed into a vector $[h 0$ ] by multiplication by a $2 \times 2$ invertible matrix.
Now we give an example of how the elementary local-global machinery no. 1 (described on p. 199) is used.

## 数学代写|交换代数代写Commutative Algebra代考|Bezout Rings

$$\forall a, b \exists u, v, g, a_1, b_1 \quad\left(a u+b v=g, a=g a_1, b=g b_1\right) .$$

$$\forall x, y \in \mathbf{A} \quad x+y \in \mathbf{A}^{\times} \Longrightarrow\left(x \in \mathbf{A}^{\times} \text {or } y \in \mathbf{A}^{\times}\right) .$$

$$\forall x \in \mathbf{A} \quad x \in \mathbf{A}^{\times} \text {or } 1-x \in \mathbf{A}^{\times} .$$

## 数学代写|交换代数代写Commutative Algebra代考|Finitely Presented Modules over Valuation Rings

$\mathbf{A}^{m \times n}$的每个矩阵基本等价于史密斯形式的矩阵。