Rings - Basic Definitions and Examples

Definition: A ring is a set $R$ with two binary operations $+$ (addition) and $\times$ (multiplication) such that

1. $R$ is an abelian group under the addition operation.
2. Multiplication is associative.
3. The distributive law holds: $a\times(b+c)=a\times b+a\times c$ and $(b+c)\times a=b\times a+c\times a$.
4. If multiplication is commutative, then $R$ is called a commutative ring.
5. If there is an identity element $1$ for multiplication, then $R$ is said to be a ring with identity or with unity.

The properties of arithmetic you expect hold. For example, if $-a$ is the additive inverse of $a$, then $(-a)(-b)=ab$. The identity element, if it exists, is unique. See DF, Proposition 1 on page 226.

Definitions: Let $R$ be a ring with unity and assume $1\not=0$.

1. A unit $x$ in $R$ is an element with a multiplicative inverse, so that there is $y$ such that $xy=yx=1$. The set $R^{*}$ of units in a ring form a group.
2. A zero-divisor in $R$ is a non-zero element $x$ such that there is $y\in R$ with $xy=0$ or $yx=0$.
3. If every non-zero element in $R$ is a unit, then $R$ is called a skew-field or a division ring.
4. A commutative skew-field is called a field.
5. A commutative ring with no zero divisors is called a domain or an integral domain.

Examples:

1. The rational numbers $\mathbb{Q}$, real numbers $\mathbb{R}$, and complex numbers $\mathbb{C}$ are all examples of fields.
2. The integers $\Z$ are a commutative ring with unity. It is also an integral domain, but not a field.
3. For each $n\ge 2$, the groups $\Zn{n}$ are actually commutative rings with unity.
4. If $p$ is prime, then $\Zn{p}$ is a field.
5. If $n$ is composite, then $\Zn{n}$ is not a domain. Its units $(\Zn{n})^*$ are the multiplicative group of elements relatively prime to $n$.
6. Let $R=C([0,1],\R)$ be the space of continuous real-valued functions on $[0,1]$. $R$ is a commutative ring with unity; its units are the non-vanishing functions. If $f$ vanishes at a point, then $f$ is not a unit, but it is also not a zero divisor.
7. The Gaussian integers $\Z[i]$ are an integral domain. So is $\Z[\sqrt{2}]$.
8. If $R$ is a ring, then the $n\times n$ matrices over $R$ are a ring $M_{n}(R)$. If $R$ has a unit element, so does $M_{n}(R)$.
9. If $R$ is a commutative ring with unity, then $R[x]$, the polynomials over $R$, are a commutative ring with unity.
10. If $R$ is any commutative ring with unity and $G$ is a finite group, then the group ring $R[G]$ consists of functions on $G$ with multiplication by “convolution” \((a*b)(g)=\sum_{h}a(h)b(gh^{-1})\)
11. The real quaternion ring $\mathbb{H}$ consists of sums $a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$ with multiplication using the usual quaternion rules. $\mathbb{H}$ is a division ring.

Proposition: A finite integral domain is a field.

Ring homomorphisms, ideals, and quotients

Definition: An subset $I$ of a ring $R$ is a left ideal if it is a subring of $R$ (meaning it is closed under addition and multiplication) such that, for all $a\in R$, $aI=\lbrace ax: x\in I\rbrace$ is contained in $I$. It is a right ideal if $Ia\subset I$ for all $a\in R$. It is an ideal if it is both a left and right ideal.

Definition: A map $\phi:R\to S$ of rings is a homomorphism if $\phi(a+b)=\phi(a)+\phi(b)$ and $\phi(ab)=\phi(a)\phi(b)$.

Lemma: The kernel of a ring homomorphism (the elements that map to zero) form an ideal of $R$. The image is a subring (i.e. closed under addition and multiplication).

Definition: If $R$ is a ring and $I$ is an ideal, then the quotient $R/I$ consists of elements of the quotient group $R/I$ with multiplication defined by $(a+I)(b+I)=(ab+I)$. This is well defined because $I$ is an ideal. The map $R\to R/I$ given by $a\mapsto a+I$ is a ring homomorphism with kernel $I$ called the canonical projection.

Theorem: Let $f:R\to S$ be a homomorphism of rings, let $I$ be the kernel of $f$, and let $\pi:R\to R/I$ be the canonical projection. Then there is a unique injective homomorphism $\overline{f}:R/I\to S$ such that $\overline{f}\circ\pi=f$.

\[\begin{xy} \xymatrix { R\ar[rd]^{f}\ar[d]^{\pi} & \\ R/I\ar[r]_{\overline{f}} & S\\ } \end{xy}\]

Properties of ideals (in rings with unity)

Assume that $R$ has an identity element for these definitions and theorems.

Definition: If $A\subset R$ is a subset, then $RA$ is the smallest left ideal containing $A$, $AR$ is the smallest right ideal containing $A$, and $RAR$ is the smallest ideal containing $A$.

a. If $I$ is generated by finitely many elements, it is called finitely generated.

b. If it is generated by one element, it is called principal. If $R$ is commutative, a principal ideal is the multiples $aR$ of a given element $a\in R$.

c. $I$ is called maximal if the only ideals of $R$ containing $I$ are $I$ and $R$.

d. If $R$ is commutative, then an ideal $P$ is called prime if $P\not=R$ and $ab\in P$ implies either $a\in P$ or $b\in P$.

Lemma: Suppose $R$ is commutative.

• $P$ is prime if and only if $R/P$ is a domain.
• $P$ is maximal if and only if $R/P$ is a field.

Theorem: Every ideal $I\subset R$ is contained in a maximal ideal.

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