Algebra Notes
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04-02-2015, 11:29 AM
Post: #1
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Algebra Notes
1 Background 1.1 Rings Ring 1.1 Definition A Ring $R\;(R,+,\cdot)$ is always assumed commutative with $1$: $\quad\bullet \;(R,+)$ is an abelian group (write $0$ for the additive identity), $\quad\bullet \;(R,\cdot)$ is associative, commutative, and $\exists 1\in R\,\forall a\in R\;(1\cdot a = a = a\cdot 1)$ $\quad\bullet \;\forall a,b,c\in R\;(a\cdot (b+c) = a\cdot b + a\cdot c)$ (Distributive law) $\quad\bullet \;0\ne 1$ (additive and multiplicative identities are different) Write $ab$ for $a\cdot b$. Integral domain 1.1.2 Definition An integral domain is a ring s.t. $\small(ab = 0)\implies (a = 0)\vee (b=0)$. Field 1.1.3 Definition A field is a ring $F$ such that $\forall b\in F^{\times}\;\exists b^{-1}\in F^{\times}\;(bb^{-1} = 1)$ $\qquad F^{\times}:= F-\{0\}$ is the multiplicative group of $F.$ (Write $a/b$ or $\dfrac{a}{b}$ for $ab^{-1}$). $\qquad$We reserve letter $F$ for a field unless otherwise stated. 1.1.4 Remark A field is an integral domain. 1.1.5 Examples $\mathbb{Z}$ is an integral domain while $\mathbb{Q,\,R,\,C}$ are fields. Subring, Subfield and Ideal 1.1.6 Definitions $S$ is a subring of a ring $R\;$ iff $\;(1\in S\cdot S = S\le_+ R)$. $\quad S\,$is a subfield of a field $F\,$ iff $(S\subset_R F)\wedge(S-\{0\}\le_{\LARGE\mathbf{\cdot}} F^{\times})$. $\quad I\,$is an ideal in a ring $R$ iff $(1\not\in I\le_+ R)\wedge (RI\subset I)$. 1.1.7 Lemma If $I$ is an ideal in a ring $R$, then $R/I = \{r+I\mid r\in R\}$ $\quad\color{grey}{\small{(r+I = \{r+i\mid i\in I\})}}\quad$form a (quotient) ring with $\;\; +,\,\cdot$ given by $\quad (r+I) + (s+I) = (r+s) +I,\;(r+I)(s+I) = rs + I.\;$ 1.1.8 Lemma $\quad\mathbb{Z}_n :=\mathbb{Z}/n\mathbb{Z}$ is the ring of integers modulo $n$ $\quad\small(\in\mathbb{N}-\{1\},\;\mathbb{Z}_0 = \mathbb{Z})$and $\mathbb{Z}_n$ is a field iff $n$ is a prime number. $\quad$The elements of $\mathbb{Z}_n$ will be denoted by $0,1,\ldots,n-1$ or $\bar{0},\,\bar{1},\ldots,\overline{n-1}$ $\quad$instead of $...,k + n\mathbb{Z},..$ 1.1.9 Definition $\phi:R\to S$ is a homomorphism of rings if $\qquad(\phi(1) = 1)\wedge \forall a,b\in R\;(\phi(a+b) = \phi(a) +\phi(b),\,\phi(ab) = \phi(a)\phi(b))$ $\qquad\phi$ is monomorphism,epimorphism, or isomorphism if $\phi$ is in additionally $\qquad$injective, surjective,or bijective, respectively. 1.1.10 Proposition If $\phi:R\to S$ is a ring homomorphism, then $\text{Im}(\phi)$ is $\qquad$a subring of $S$ and $\ker(\phi)$ is an ideal in $R$, and $\bar{\phi}:{\small{\dfrac{R}{\ker(\phi)}}}\to \text{Im}(\phi)$ $\qquad\color{grey}{\small{(\bar{\phi}(r+\ker(\phi)) = \phi( r))}}$ is a ring isomorphism. |
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Messages In This Thread |
Algebra Notes - elim - 04-02-2015 11:29 AM
1.2 Prime subfields - elim - 04-02-2015, 02:02 PM
1.3 Fields of fractions - elim - 04-02-2015, 04:40 PM
1.4 Polynomial rings - elim - 04-02-2015, 05:11 PM
1.5 Polynomial rings over fields - elim - 04-03-2015, 04:40 PM
RE: Algebra Notes - elim - 04-04-2015, 08:22 PM
2 Field Extensions - elim - 04-06-2015, 07:24 AM
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