Rings

Why

We generalize the algebraic structure of addition and multiplication over the integers.¹

Definition

A ring (or ring with identity) $(R, f, g)$ is a set $A$ and two binary operations on $R$ satisfying the following set of conditions.

(A) (i) $f$ is associative. (ii) $f$ is commutative, (iii) $A$ has an identity element for $f$ (i.e., there is $e \in R$ with $f(r, e) = f(e, r) = r$ for all $r \in R$ (iv) $R$ has inverse elements for $f$ (i.e., for any $r \in R$, there is $\tilde{r} $ satisfying $f(r,\tilde{r}) = f(\tilde{r}, r) = e$)

(B) (i) $g$ is associative; (ii) $R$ has an identity element for $g$ (i.e., for any $r \in R$, there is $\tilde{e} \in A$ satisfying $g(r, \tilde{e}) = g(\tilde{e},r) = r$)

(C) (i) $g$ left distributes:

\[ g(f(x, y), \alpha ) = f(g(\alpha ,x), g(\alpha ,y)) \quad \text{for all } x, y, \alpha \in R \]

Conditions (A) concern $f$, conditions (B) concern $g$, and conditions (C) relate the two.

Clearly, $\Z $ with addition and multiplication is a ring. The element referred to in (A.2) is $0 \in \Z $, so we refer to this element in any ring as the additive identity. That referred to (A.3) is $1 \in \Z $, so we refer to this element in any ring as the multiplicative identity. We refer to the elements mentioned in (A.4) as additive inverses. We call to $f$ ring addition and $g$ ring multiplication.

A ring which for which multiplication is commutative is called a commutative ring. Note that a ring is always commutative with respect to addition, here the term commutative refers to multiplication. A ring for which there are inverse elements, excepting 0, is called a division ring).

Of course, the integers form a ring with the usual notion of addition and multiplication. For another trivial example, consider $\set{0}$ with $0+0 = 0$ and $0\cdot 0 = 0$; this is called the zero ring (any ring isomorphic to this one is called a trivial ring or zero ring).

Notation

The notation commonly adopted in discussing rings relies on analogy with the set of integers $\Z $. We denote the ring addition by $+$ and ring multiplication by $\cdot $. Moreover, we denote the ring’s additive identity by $0$ and the ring’s multiplicative identity by $1$. Finally, we denote the additive inverse of $a \in A$ by $-a$.

Rewriting the conditions (A), (B), (C) in this notation gives familiar-looking relations, from when the objects involved were integers. (A) (1) $a+(b + c) = (a+b)+c$; (2) $a+b = b+a$; (3) $a + 0 = 0 + a = a$; (4) $a + (-a) = 0$. (B) (1) $a(bc) = (ab)c$; (2) $1a = a1 = a$. (C) (1) $(a+b)c = ac + bc$; (2) $c(a+b) = ca + cb$.

Immediate consequences

We need not require that $0x = 0$, because we can deduce it:

\[ 0x + x = (0 + 1)x = 1x = x. \]

1. Future editions will likely modify this sheet, and give a genetic treatment involving the solution of polynomial equations by Galois. ↩︎