Algebra Notes

[elim]

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.