全部书籍

1 Primary & Middle School Mathematics1–9 年级在线 2 High School Mathematics10–12 年级在线 3 University Mathematics — Year 1本科一年级在线 4 University Mathematics — Year 2本科二年级在线 5 University Mathematics — Year 3本科三年级在线

1 Primary & Middle School Physics1–9 年级在线 2 High School Physics10–12 年级在线 3 University Physics — Year 1本科一年级在线 4 University Physics — Year 2本科二年级在线 5 University Physics — Year 3本科三年级在线

1 School Chemistry — Grades 1 to 121–12 年级在线 2 University Chemistry — Year 1本科一年级在线 3 University Chemistry — Year 2本科二年级在线 4 University Chemistry — Year 3本科三年级在线

1 Primary & Middle School Biology1–9 年级在线 2 High School Biology10–12 年级在线 3 University Biology — Year 1本科一年级在线 4 University Biology — Year 2本科二年级在线 5 University Biology — Year 3本科三年级在线

1 Primary & Middle School Earth and Space Sciences1–9 年级在线 2 High School Earth and Space Sciences10–12 年级在线 3 University Earth and Space Sciences — Year 1本科一年级在线 4 University Earth and Space Sciences — Year 2本科二年级在线 5 University Earth and Space Sciences — Year 3本科三年级在线

1 Markets I: The Ecosystem and Exchange-Traded Markets市场在线 2 Markets II: Rates, FX and Credit市场在线 3 Markets III: Commodities, Energy and Crypto市场在线 4 Quantitative Methods方法在线 5 Derivatives and Volatility衍生品在线 6 Rates, Credit, XVA and Risk利率、信用与风险在线 7 Research Craft: Predictors, Backtests, Measurement, Portfolios研究在线 8 Strategies I: Equities and Futures策略在线 9 Strategies II: Volatility, Relative Value, Macro and the Bank Desks策略在线 10 Microstructure and Execution交易执行在线 11 Market Making and High-Frequency Trading做市在线 12 Machine Learning for Markets机器学习在线 13 Low-Latency Software技术在线 14 Networks, Hardware and Trading Infrastructure技术在线 15 Research, Data and Risk Platforms技术在线 16 The Desk and the Firm公司运营在线 17 The Industry: Firms, Roles and Careers职业发展在线 18 Interviews I: Maths, Probability and Speed职业发展在线 19 Interviews II: Markets, Trading Games and the Final Round职业发展在线 20 Interviews III: Code职业发展在线

全部游戏 Math Fuse Word Reactor Atlas Dash

应用关于

Coach

English Français Español Nederlands Português हिन्दी العربية Bahasa Indonesia 简体中文 বাংলা

登录开始阅读

书籍暂未推出简体中文版——当前显示英文版。

Mathematics · 术语表

什么是 module？

Definition 3.1 University Mathematics — Year 3 · 第 3 章 — Modules over a Principal Ideal Domain

An $A$ -module is an abelian group $(M, +)$ with a scalar multiplication $A \times M \to M$ satisfying the vector-space axioms: $a(x + y) = ax + ay$ , $(a + b)x = ax + bx$ , $(ab)x = a(bx)$ , $1x = x$ . Submodules, quotients $M/N$ , morphisms ( $A$ -linear maps), direct sums $\bigoplus_i M_i$ , and the isomorphism theorems are defined and proved word for word as for vector spaces and abelian groups; in particular $M/\ker f \cong \operatorname{im} f$ for a morphism $f$ .

示例

Example 3.2

The three motivating cases.

$A = K$ a field: modules are vector spaces.
$A = \Z$ : modules are exactly abelian groups ( $nx$ is forced to be $x + \dots + x$ ), submodules are subgroups.
$A = K[X]$ : a module is a $K$ -vector space $V$ together with the $K$ -linear map $u\colon x \mapsto X\cdot x$ — conversely, every pair $(V, u)$ with $u \in \mathcal L(V)$ becomes a $K[X]$ -module by $P \cdot x = P(u)(x)$ . The submodules are precisely the $u$ -stable subspaces.

An ideal of $A$ is exactly a submodule of $A$ ; a quotient ring $A/I$ is an $A$ -module. Unlike vector spaces, modules can have torsion: in $\Z/6\Z$ , the element $\bar 3 \ne 0$ is killed by $2 \neq 0$ .

在原文中阅读 →