From 6968afce16499dda67a1eb603383655d802d5b89 Mon Sep 17 00:00:00 2001 From: semapheur Date: Tue, 7 May 2024 06:56:53 +0200 Subject: [PATCH] Deploy --- content/notes/math/algebra.mdx | 77 +++++++++++++++++++++++++++------- 1 file changed, 61 insertions(+), 16 deletions(-) diff --git a/content/notes/math/algebra.mdx b/content/notes/math/algebra.mdx index 7ecba89..186e120 100644 --- a/content/notes/math/algebra.mdx +++ b/content/notes/math/algebra.mdx @@ -4,7 +4,7 @@ subject: 'Mathematics' showToc: true --- -# Category +# Category theory - -A ring is a non-empty set $R$ together with two binary (closed) operations, $+: R \times R \to R$ (addition) and $\cdot: R\times R\to R$, with the properties -- $(R, +)$ is an abelian group -- $(R, \cdot)$ is a monoid -- Multiplication is distributive related to addition, i.e. for all $a,b,c\in R$ - - $(a + b)c = ac + bc$ - - $c(a + b) = ca + cb$ - -A ring is commutative if $ab = ba$ for all $a,b\in R$. If a ring $R$ contains an element $e$ such that $ae = ea = a$ for all $a\in R$, we say that $R$ is a ring with identity. - - A field $\mathbb{F}$ is a set containing $1 \neq 0$ equipped with the binary operations $+$ (addition) and $\cdot$ (multiplication) such that - $(\mathbb{F}, +)$ is an abelian group - $(\mathbb{F}\setminus\Set{0}, \cdot)$ is an abelian group -- Multiplication is distributive related to addition, i.e. for all $a,b,c\in\mathbb{F}$ - - $a(b + c) = ab + ac$ - - $(a + b)c = ac + bc$ +- Addition and multiplication are related by distributivity, i.e. for all $a,b,c\in\mathbb{F}$ + - $a(b + c) = ab + ac$ + - $(a + b)c = ac + bc$ @@ -289,4 +278,60 @@ for which the following properties hold: - $A$ is a vector space over $\mathbb{F}$ under addition and scalar multiplication - $A$ is a ring under addition and multiplication - if $r\in\mathbb{F}$ and $a,b\in A$ then $r(ab) = (ra)b = a(rb)$ (scalar compatibility) - \ No newline at end of file + + +# Ring theory + + +A ring is a non-empty set $R$ equipped with two binary (closed) operations, +- $+: R \times R \ni (a,b)\mapsto a + b \in R$ **(addition)** +- $\cdot: R\times R \ni (a,b)\mapsto ab \in R$ **(multiplication)*** + +with the properties +- $(R, +)$ is an abelian group +- $(R, \cdot)$ is a monoid +- Addition and multiplication are related by distributity, i.e. for all $a,b,c\in R$ + - $(a + b)c = ac + bc$ + - $c(a + b) = ca + cb$ + +A ring is commutative if $ab = ba$ for all $a,b\in R$. If a ring $R$ contains an element $e$ such that $ae = ea = a$ for all $a\in R$, we say that $R$ is a ring with identity. + + + +Let $a,b$ be in a ring $R$. If $a \neq 0$ and $b \neq 0$ but $ab = 0$, then $a$ and $b$ are *zero divisiors*. If $ab = ba = 1$, we say the $a$ is a *unit*, or that $a$ is invertible. + + + +An *integral domain* is a commutative ring with no zero divisor. A *division ring* or *skew field* is a ring in which every non-zero element $a$ has an inverse $a^{-1}$. A *field* is a commutative ring in which every non-zero element is invertible. + + + +The *characteristic* of a ring $R$, denoted $\operatorname{char}(R)$, is the smallest positive integer such that + +$$ + n\cdot 1 = \underbrace{1 + \cdots + 1}_{n \text{ times}} = 0 +$$ + +If there is no such positive integer, then $\operatorname{char}(R) = 0$. + + + +A subring of a ring $R$ is a subset $S$ of $R$ forming a ring under the operations of addition and multiplication defined in $R$. + + + +Let $R, S$ be two rings. A map $f:R\to S$ is a ring homomorphism if it satisfies for $a,b\in R$ +1. $f(a + b) = f(a) + f(b)$ (group homomorphism) +2. $f(ab) = f(a)f(b)$ +3. $f(1_R) = 1_S$ + + + +Let $\mathcal{I}$ be a subset of ring $R$. An additive subgroup of $R$ is called +- a *left ideal* of $R$ if $ra\in\mathcal{I}$ for $a\in\mathcal{I}$ and $r\in R$ +- a *right ideal* of $R$ if $ar\in\mathcal{I}$ for $a\in\mathcal{I}$ and $r\in R$ + +A left ideal that is also a right ideal, is called a *(two-sided) ideal* of $R$. An ideal is *proper* if $\mathcal{I} \neq R$, and it is *non-trivial* if in addition $\mathcal{I}\neq 0$. + + + \ No newline at end of file