Abstract Algebra/Rings, ideals, ring homomorphisms

Basic definitions

Definition 10.1:

A ring is a set $R$ together with two binary operations $+:R\times R\to R$ and $\cdot :R\times R\to R$ and two special elements, the unit $1$ and the zero $0$ , such that:

$R$ is an abelian group with respect to $+$ with neutral element $0$ .
$R$ is a monoid (that is, a group without inversion) with respect to $\cdot$ with neutral element $1$ .
The distributive laws hold: $a\cdot (b+c)=a\cdot b+a\cdot c$ , $(b+c)\cdot a=b\cdot a+c\cdot a$ .

Examples 10.2:

The whole numbers $\mathbb {Z}$ with respect to usual addition and multiplication are a ring.
Every field is a ring.
If $R$ is a ring, then all polynomials over $R$ form a ring. This example will be explained later in the section on polynomial rings.

Definition 10.3:

Let $R$ be a ring. A left ideal of $R$ is a subset $I\subseteq R$ such that the following two things hold:

$(I,+)$ is a subgroup of $(R,+)$ .
$\forall r\in R:rI\subseteq I$ , where $rI=\{ri|i\in I\}$ (closedness by left multiplication).

Replacing closedness by left multiplication by closedness by right multiplication, we can define right ideals, and then both-sided ideals. If $I$ is a both-sided ideal of $R$ , we write $I\leq R$ .

Residue class rings

Definition and theorem 10.4:

Let $R$ be a ring, and $I\leq R$ . Then we define a relation $\sim _{I}$ on $R$ as follows:

a\sim _{I}b:\Leftrightarrow a-b\in I

This relation is an equivalence relation, and an equivalence class $[a]$ shall be denoted by $a+I$ for $a\in R$ . If we define an addition

(a+I)+(b+I):=(a+b)+I

and a multiplication

(a+I)\cdot (b+I):=a\cdot b+I

then these two are well-defined (i. e. independent of the choice of the representatives $a$ and $b$ ) and turn $R/I$ into a ring, called the residue class ring with respect to the ideal $I$ .

Proof:

First, we check that $\sim _{I}$ is an equivalence relation.

Reflexiveness: $a-a=0\in I$ since $I$ is an additive subgroup.
Symmetry: $a-b\in I\Leftrightarrow -(a-b)\in I$ since inverses are in the subgroup.
Transitivity: Let $a-b\in I$ and $b-c\in I$ . Then $a-c=a-b+(b-c)\in I$ , since a subgroup is closed under the group operation.

Then we check that addition and multiplication are well-defined. Let $a+I=a'+I$ and $b+I=b'+I$ . Then

a+b-(a'+b')=a+b-(a+i+b+j)=-i-j\in I

for certain

i,j\in I

Furthermore,

a\cdot b-a'\cdot b'=a\cdot b-a\cdot b-a\cdot j-i\cdot b-i\cdot b

for these same $i,j\in I$ ; this is in $I$ by closedness by left and right multiplication.

The ring axioms directly carry over from the old ring $R$ .

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Ring homomorphisms

Definition 10.5:

Let $R,S$ be rings. A ring homomorphism between the two is a map

\varphi :R\to S

such that:

For all $a,b\in R$ $\varphi (a+b)=\varphi (a)+\varphi (b)$ and $\varphi (a\cdot b)=\varphi (a)\cdot \varphi (b)$ .
$\varphi (1_{R})=1_{S}$ ( $1_{R}$ is the unit of $R$ and $1_{S}$ of $S$ ).

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