Rings

A nonempty set \(R\) is a ring if it has two closed binary operations, addition and multiplication, satisfying the following conditions.

1. \(a + b = b + a\) for \(a, b \in R\).

2. \((a + b) + c = a + ( b + c)\) for \(a, b, c \in R\).

3. There is an element \(0\) in \(R\) such that \(a + 0 = a\) for all \(a \in R\).

4. For every element \(a \in R\), there exists an element \(-a\) in \(R\) such that \(a + (-a) = 0\).

5. \((ab) c = a ( b c)\) for \(a, b, c \in R\).

6. For \(a, b, c \in R\), \[\begin{aligned}a( b + c)&= ab +ac \\ (a + b)c & = ac + bc\end{aligned}\].

This last condition, the distributive axiom, relates the binary operations of addition and multiplication. Notice that the first four axioms simply require that a ring be an abelian group under addition, so we could also have defined a ring to be an abelian group \((R, +)\) together with a second binary operation satisfying the fifth and sixth conditions given above.

If there is an element \(1 \in R\) such that \(1 \neq 0\) and \(1a = a1 = a\) for each element \(a \in R\), we say that \(R\) is a ring with unity or identity. A ring \(R\) for which \(ab = ba\) for all \(a, b\) in \(R\) is called a commutative ring. A commutative ring \(R\) with identity is called an integral domain if, for every \(a, b \in R\) such that \(ab = 0\), either \(a = 0\) or \(b = 0\). A division ring is a ring \(R\), with an identity, in which every nonzero element in \(R\) is a unit; that is, for each \(a \in R\) with \(a \neq 0\), there exists a unique element \(a^{-1}\) such that \(a^{-1} a = a a^{-1} = 1\). A commutative division ring is called a field. The relationship among rings, integral domains, division rings, and fields is shown in .

Example

As we have mentioned previously, the integers form a ring. In fact, \({\mathbb Z}\) is an integral domain. Certainly if \(a b = 0\) for two integers \(a\) and \(b\), either \(a=0\) or \(b=0\). However, \({\mathbb Z}\) is not a field. There is no integer that is the multiplicative inverse of \(2\), since \(1/2\) is not an integer. The only integers with multiplicative inverses are \(1\) and \(-1\).

A nonzero element \(a\) in a commutative ring \(R\) is called a zero divisor if there is a nonzero element \(b\) in \(R\) such that \(ab = 0\). In the previous example, \(3\) and \(4\) are zero divisors in \({\mathbb Z}_{12}\).

Condensed — the full section is in Judson, Abstract Algebra: Theory and Applications.