Difference between revisions of "817 - Algebra"
Kfagerstrom (talk | contribs) (→Ideals) |
Kfagerstrom (talk | contribs) |
||
| (10 intermediate revisions by the same user not shown) | |||
| Line 6: | Line 6: | ||
====[[Sylow Theory]]==== | ====[[Sylow Theory]]==== | ||
| − | ====Semi-Direct Product ==== | + | |
| + | |||
| + | Let G be a finite group and p a prime. Write the order of <math> G</math> as | ||
| + | <math> |G| = p^em </math> where <math>p \not| m</math>. A Sylow p-subgroup of <math> G</math> is a subgroup <math> H \leq G </math>such that <math> |H| = p^e</math>. That is, a Sylow p-subgroup of <math>G </math> is a subgroup whose order is the highest conceivable power of <math>p </math> according to Lagrange’s Theorem. | ||
| + | We set <math> \text{Syl}_p(G)</math> to be the collection of all Sylow p-subgroups of <math>G </math> and <math>n_p = | \text{Syl}_p(G)| </math> to be the number of Sylow p-subgroups. | ||
| + | |||
| + | ====[[Semi-Direct Product]] ==== | ||
| + | |||
====Quotient Groups==== | ====Quotient Groups==== | ||
====Isomorphism Theorems ==== | ====Isomorphism Theorems ==== | ||
| − | + | :''see also'' [[Isomorphism Theorems]] | |
==Rings== | ==Rings== | ||
| Line 20: | Line 27: | ||
===Ideals === | ===Ideals === | ||
| − | A subring <math> A </math> of a ring <math>R </math> is an ''ideal'' in <math>R </math>, if <math>aR </math> and <math>Ra </math> are subsets of <math> A</math> for every <math>a \in A </math>, and <math>( | + | A subring <math> A </math> of a ring <math>R </math> is an ''ideal'' in <math>R </math>, if <math>aR </math> and <math>Ra </math> are subsets of <math> A</math> for every <math>a \in A </math>, and <math>(A,+) </math> is an subgroup of <math>(R,+) </math>. |
| Line 28: | Line 35: | ||
| − | ====Prime Ideals ==== | + | ====[[Prime Ideals]] ==== |
A ''prime ideal'' of a ring <math>R </math> is proper ideal <math>P </math> such that whenever <math>xy\in P </math> for <math>x,y \in R </math>, <math>x\in P </math> or <math>y \in P </math>. </br> | A ''prime ideal'' of a ring <math>R </math> is proper ideal <math>P </math> such that whenever <math>xy\in P </math> for <math>x,y \in R </math>, <math>x\in P </math> or <math>y \in P </math>. </br> | ||
| − | + | An ideal <math>P </math> is prime if and only if <math> R\backslash P</math> is closed under multiplication. | |
| + | </br> | ||
| + | Maximal implies prime but not conversely. | ||
===Named Rings=== | ===Named Rings=== | ||
| + | |||
| + | ====[[Integral Domains]]==== | ||
| + | An ''Integral Domain'', often just called domain, is a commutative ring <math>R </math>, with <math>1\neq 0 </math> and has no zero divisors. | ||
==== Euclidian Domains (EDs) ==== | ==== Euclidian Domains (EDs) ==== | ||
A ''Euclidean Domain'' (ED) is a domain <math>R </math> together with a function <math>N: R \to \mathbb{Z}_{\geq 0} </math> such that <math>N(0)=0 </math> and the following property holds: for any two elements <math> a,b \in R </math> with <math>b\neq 0 </math>, there are elements <math> q,r \in R</math> such that <math> a=qb+r </math> and either <math>r=0 </math> or <math>N(r)<N(b) </math> | A ''Euclidean Domain'' (ED) is a domain <math>R </math> together with a function <math>N: R \to \mathbb{Z}_{\geq 0} </math> such that <math>N(0)=0 </math> and the following property holds: for any two elements <math> a,b \in R </math> with <math>b\neq 0 </math>, there are elements <math> q,r \in R</math> such that <math> a=qb+r </math> and either <math>r=0 </math> or <math>N(r)<N(b) </math> | ||
| − | ====Principle Ideal Domains (PIDs) ==== | + | ====[[Principle Ideal Domains]] (PIDs) ==== |
| + | A ''Principal Ideal Domain'' (PID) is a | ||
| + | domain, <math>R</math> with the property that every ideal is principal, i.e., for each ideal <math>I</math>, we have | ||
| + | <math>I = (a) </math> for some <math>a \in R</math>. | ||
| + | |||
====Unique Factorization Domains (UFDs) ==== | ====Unique Factorization Domains (UFDs) ==== | ||
| + | |||
| + | A ''Unique Factorization Domain'' (UFD) is an integral domain such that every element <math>r \in R</math> that is non-zero and not a unit can | ||
| + | be written as a finite product <math>r = p_1 \cdots p_n </math> | ||
| + | of (not necessarily distinct) irreducible elements <math>p_1,..., p_n</math> of <math>R</math> in a way that is unique up to ordering and associates. That is, if <math>r = q_1 · · · q_m</math> also holds with each <math>q_i</math> irreducible, then <math>m = n</math> and there is a permutation <math>\sigma </math> such that, for all <math>i</math>, we have <math>p_i</math> and <math>q\sigma(i)</math> are associates. | ||
Latest revision as of 20:44, 17 January 2023
Groups
Theorems
Topics in Group Theory
Sylow Theory
Let G be a finite group and p a prime. Write the order of \( G\) as \( |G| = p^em \) where \(p \not| m\). A Sylow p-subgroup of \( G\) is a subgroup \( H \leq G \)such that \( |H| = p^e\). That is, a Sylow p-subgroup of \(G \) is a subgroup whose order is the highest conceivable power of \(p \) according to Lagrange’s Theorem. We set \( \text{Syl}_p(G)\) to be the collection of all Sylow p-subgroups of \(G \) and \(n_p = | \text{Syl}_p(G)| \) to be the number of Sylow p-subgroups.
Semi-Direct Product
Quotient Groups
Isomorphism Theorems
- see also Isomorphism Theorems
Rings
Definition: A ring is a set \(R\) with two binary operation \( +\) and \(\cdot\) satisfying:
- \( (R,+)\) is an abelian group (with identity 0)
- \( (R,\cdot) \) is a semigroup
- \(\cdot \) is distributive over \( + \) (on both sides)
Ideals
A subring \( A \) of a ring \(R \) is an ideal in \(R \), if \(aR \) and \(Ra \) are subsets of \( A\) for every \(a \in A \), and \((A,+) \) is an subgroup of \((R,+) \).
Maximal Ideals
A maximal ideal of a ring \(R \) is proper ideal \(M \) such that the only ideals in \( R\) containing \(M \) are \( M\) and \(R \).
Prime Ideals
A prime ideal of a ring \(R \) is proper ideal \(P \) such that whenever \(xy\in P \) for \(x,y \in R \), \(x\in P \) or \(y \in P \).
An ideal \(P \) is prime if and only if \( R\backslash P\) is closed under multiplication.
Maximal implies prime but not conversely.
Named Rings
Integral Domains
An Integral Domain, often just called domain, is a commutative ring \(R \), with \(1\neq 0 \) and has no zero divisors.
Euclidian Domains (EDs)
A Euclidean Domain (ED) is a domain \(R \) together with a function \(N: R \to \mathbb{Z}_{\geq 0} \) such that \(N(0)=0 \) and the following property holds: for any two elements \( a,b \in R \) with \(b\neq 0 \), there are elements \( q,r \in R\) such that \( a=qb+r \) and either \(r=0 \) or \(N(r)<N(b) \)
Principle Ideal Domains (PIDs)
A Principal Ideal Domain (PID) is a domain, \(R\) with the property that every ideal is principal, i.e., for each ideal \(I\), we have \(I = (a) \) for some \(a \in R\).
Unique Factorization Domains (UFDs)
A Unique Factorization Domain (UFD) is an integral domain such that every element \(r \in R\) that is non-zero and not a unit can be written as a finite product \(r = p_1 \cdots p_n \) of (not necessarily distinct) irreducible elements \(p_1,..., p_n\) of \(R\) in a way that is unique up to ordering and associates. That is, if \(r = q_1 · · · q_m\) also holds with each \(q_i\) irreducible, then \(m = n\) and there is a permutation \(\sigma \) such that, for all \(i\), we have \(p_i\) and \(q\sigma(i)\) are associates.
