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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|交换代数代写commutative algebra代考|Exterior Powers of a Module

We now introduce several constructive notions relating to the equality test and the membership test.

A set $E$ is well defined when we have indicated how to construct its elements and when we have constructed an equivalence relation which defines the equality of two elements in a set. We denote by $x=y$ the equality in $E$, or $x={ }_{E} y$ if necessary. The set $E$ is called discrete when the following axiom holds
$$\forall x, y \in E \quad x=y \text { or } \neg(x=y)$$
Classically, every set is discrete, as the “or” present in the definition is understood in an abstract manner. Constructively, this same “or”‘ is understood according to the usual language’s meaning: at least one of the two alternatives must occur. It is thus an “or” of an algorithmic nature. In short, a set is discrete if we have a test for the equality of two arbitrary elements of this set.

If we want to be more precise and explain in detail what comprises an equality test in the set $E$, we will say that it is a construction which, from two given elements of $E$, provides a “yes” or “no” answer to the posed question (are these elements equal?). However, we could not go into much further detail. In constructive mathematics the notions of integers and of construction are basic concepts. They can be explained and commented on, but not strictly speaking “defined.” The constructive meaning of the “or” and that of the “there exists” are as such directly dependent of the notion of construction, ${ }^{6}$ which we do not attempt to define.
A discrete field is simply a ring where the following axiom is satisfied:
$$\forall x \in \mathbf{A} \quad x=0 \text { or } x \in \mathbf{A}^{\times}$$
The trivial ring is a discrete field.

## 数学代写|交换代数代写commutative algebra代考|Strongly Discrete Coherent Rings and Modules

A ring (resp. a module) is said to be strongly discrete when the finitely generated ideals (resp. the finitely generated submodules) are detachable, i.e. if the quotients by the finitely generated ideals (resp. by the finitely generated submodules) are discrete.
This means that we have a test for deciding whether a linear equation $L X=c$ has a solution or not, and by computing one in the affirmative case.

A key result in constructive algebra and Computer Algebra states that $\mathbb{Z}\left[X_{1}, \ldots, X_{n}\right]$ is a strongly discrete coherent ring.

More generally, we have the following constructive version of the Hilbert theorem (see [MRR, Adams \& Loustaunau]).

If $\mathbf{A}$ is a strongly discrete Noetherian coherent ring, so is any finitely presented A-algebra.
The following proposition is proven similarly to Proposition 3.1.
3.7 Proposition Over a strongly discrete coherent module $M$, every system of linear equations $B X=C\left(B \in M^{k \times n}, C \in M^{k \times 1}, X \in \mathbf{A}^{n \times 1}\right)$ can be tested. In the affirmative case, a particular solution $X_{0}$ can be computed. Furthermore the solutions $X$ are all the elements of $X_{0}+N$ where $N$ is a finitely generated $\mathbf{A}$-submodule of $\mathbf{A}^{n \times 1}$.

# 交换代数代考

## 数学代写|交换代数代写commutative algebra代考|Exterior Powers of a Module

$$\forall x, y \in E \quad x=y \text { or } \neg(x=y)$$

$$\forall x \in \mathbf{A} \quad x=0 \text { or } x \in \mathbf{A}^{\times}$$

## 数学代写|交换代数代写commutative algebra代考|Strongly Discrete Coherent Rings and Modules

$3.7$ 关于强离散相干模块的命题 $M$ ，每个线性方程组 $B X=C\left(B \in M^{k \times n}, C \in M^{k \times 1}, X \in \mathbf{A}^{n \times 1}\right)$ 可以测试。在肯定的情况下，一个特定的解决方案 $X_{0}$ 可以计

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## MATLAB代写

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