LEC ALG3 2

Quotients

Definition 1(Definition).

Let $I\subseteq R$ and $(I,+)$ be an abelian group. $I$ is a left/right ideal if it is closed under left/right multiplication by elements of $R$.

Let $I$ be a subgroup of the abelian group $(R,+)$ of a ring $R$. Subgroups of abelian groups are automatically normal, so we have a quotient group $R/I$, whose elements are cosets $r+I$ of $I$. Further, we have a surjective group homomorphism

$$\pi :R\to R/I,r\mapsto r+I.$$

What requirements should $I$ meet, in order to have a ring structure on $R/I$, such that $\pi$ becomes a ring homomorphism? If $\pi$ is a ring homomorphism, there is only one way to define a ring structure on $R/I$:

$$(a+I)(b+I)=\pi (a)\pi (b)=\pi (ab)=ab+I.$$

Thus, there is only one sensible ring structure on $R/I$, given by $(a+I)(b+I)\equiv ab+I$. Note that if this operation is well defined, $R/I$ is a ring. When is this operation well defined?

Claim 2(Claim).

The operation $(a+I)(b+I)\equiv ab+I$ on $R/I$ is well defined iff $I$ is an ideal of $R$.

Proof.

Assume the operation is well defined, making$R/I$ into a ring, and $\pi$ into a ring homomorphism. $\ker \pi =I$. The absorption properties are easily verified: for all $a\in I$ and $r\in R$,

$$\begin{array}{r}\pi (ra)=\pi (r)\pi (a)=\pi (r)\cdot 0=0,\\ \pi (ar)=\pi (a)\pi (r)=0\cdot \pi (r)=0.\end{array}$$

Thus, $I$ is an ideal.

Conversely, assume $I$ is an ideal. Let $\alpha ,\beta \in I$.

$$\begin{array}{rl}& (a+\alpha +I)(b+\beta +I)\\ & =(a+\alpha )(b+\beta )+I\\ & =ab+\alpha b+a\beta +\alpha \beta +I\\ & =ab+I,\end{array}$$

which proves that the operation is well defined.□

Thus, $R/I$ is a ring, in such a way that the canonical projection $\pi :R\to R/I$ is a ring homomorphism, iff $I$ is an ideal of $R$.

The mapping property of quotient groups provides the scaffolding for its analogue in $\text{Ring}$: the needed (group) homomorphism exists and is unique by the group theoretic theorem; verifying it is a ring homomorphism is immediate.

Theorem 3(Aluffi (2009) III.3.8).

Let $I$ be a two-sided ideal of a ring $R$. Then for every ring homomorphism $\phi :R\to S$ such that $I\subseteq \ker \phi$ there exists a unique ring homomorphism $\tilde{\phi }:R/I\to S$ so that the diagram

commutes.

This allows for the canonical decomposition and the first isomorphism theorem for rings. The realization that the ideals of a quotient $R/I$ are in bijective correspondence with ideals of $R$ containing $I$ leads to the third isomorphism theorem:

Theorem 4(Aluffi (2009) III.3.11).

Let $I$ be an ideal of a ring $R$, and let $J$ be an ideal of $R$ containing $I$. Then $J/I$ is an ideal of $R/I$, and

$$\frac{R/I}{J/I}\cong \frac{R}{J}.$$

Warning

$J/I$ is not a ring!

Characteristic

The fact that $\mathbb{Z}$ is initial in $\text{Ring}$ prompts a natural definition. For a ring $R$, let $f:\mathbb{Z}\to R$ be the unique ring homomorphism. Then, $\ker f=n\mathbb{Z}$ for a well-defined nonnegative integer $n$ determined by $R$. This is called the characteristic of $R$.

Definition 5(Definition).

The characteristic of a ring is the smallest integer $n>0$ such that $nr=0$. If $nr\ne 0$ for all $n>0$, $\text{char}(R)=0$.

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Definition 6(Definition).

The center of a ring $R$

$$C(R)=\{c\in R|cr=rc\forall r\in R\}.$$

Exercise: this is a subring of $R$.

The entire theory of modules is based on the observation that ${\text{End}}_{\text{Ab}}(G)$ is a ring for every abelian group $G$.

Definition 7(Module).

A left $R$ module, where $R$ is a ring with $1$, is an additive abelian group $M$ with the operation $R\times M\to M$ $(r,m)\mapsto rm$ with the following axioms

1. $(r+s)m=rm+sm$
2. $r(m+n)=rm+rn$
3. $rs(m)=r(sm)$
4. $1m=m$

Ditto for right $R$ module.

Note

Let $S\subseteq R$ be a subring. Let $1_S$ and $1_R$ be the identities of $S$ and $R$ respectively. Then,

$$\begin{array}{r}{1}_S{1}_R=1_S\\ 1_S{1}_S=1_S\\ ⟹1_S(1_S-1_R)=0\end{array}$$

If $R$ is an integral domain, $1_S=1_R$ is forced. Also, if we require in our definition of ring homomorphisms for identity to be mapped to identity, the inclusion map $S\to R$ forces $1_S=1_R$. In this course, you can assume $1_S=1_R$.

If $R$ is a ring and $I$ is a two sided ideal, what are the ideals in $R/I$?

We have a natural homomorphism $\pi :R\to R/I$: $r\mapsto r+I$. If $J$ is an ideal in $R$, then $(\pi (J))(R/I)$ (the ideal generated by $\pi (J)$) in $R/I$ is an ideal in $R/I$. and is the ideal $(J+I)/I$.

[!Example]
What are the ideals in $\mathbb{Z}[x]$?

[!Definition]
Let $\phi :R\to S$ be a ring homomorphism. Let $I$ be an ideal in $R$. Consider $\phi (I)S$ (the ideal generated by $\phi (I)$). This is also called the extension of the ideal $I$ in $R$.

[!Theorem]
Let $J$ be an ideal in $S$. Define

$$I\equiv \{r\in R|\phi (r)\in J\}.$$

$I$ is an ideal in $R$.

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References

Aluffi, P. (2009). Algebra: Chapter 0. American Mathematical Society.