Difference between revisions of "817 - Algebra"
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| − | 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>. |
Revision as of 16:09, 7 December 2022
Groups
Theorems
Topics in Group Theory
Sylow Theory
Semi-Direct Product
Quotient Groups
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 \(y \in P \).
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Named Rings
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 \(N(r)<N(b) \)
