An abelian group is a set in which addition and subtraction are possible. A ring is a set where multiplication is available as well. We now make this precise.
Let $R$ be a nonempty set, and let $+$ and $\cdot$ be binary operations on $R$. The triple $(R,+,\cdot)$ is called a ring if:
- (R1) $(R,+)$ is an abelian group.
- (R2) $(R,\cdot)$ is a monoid.
- (R3) For all $a,b,c\in R$, the distributive laws hold:
$$a(b+c)=ab+ac, \qquad (a+b)c=ac+bc.$$
When the operations are clear from context, we simply write $R$ instead of $(R,+,\cdot)$.
The identity element for addition is denoted by $0_R$, while the identity element for multiplication is denoted by $1_R$. When the ring is clear, we write only $0$ and $1$.
First examples
The integers $(\mathbb Z,+,\cdot)$ form a ring. The natural numbers $\mathbb N$ do not form a ring under the usual operations because additive inverses are missing.
The polynomial ring
$$\mathbb Z[x]=\left\{\sum_{i=0}^{d}a_i x^i : d\ge 0,\ a_i\in\mathbb Z\right\}$$
is a ring under ordinary polynomial addition and multiplication.
Commutative rings
A ring $R$ is commutative when $$ab=ba$$ for every $a,b\in R$.
If $R$ and $S$ are rings, then $R\times S$ is a ring under componentwise addition and multiplication.
Both operations are defined componentwise, so associativity, distributivity, and the identity axioms follow from the corresponding properties in $R$ and $S$.
Exercises
- Verify that $2\mathbb Z$ is a ring under the usual operations. Does it have a multiplicative identity?
- Show that $M_2(\mathbb Z)$ is not commutative.
- Determine the zero and identity elements of $R\times S$.