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A ring is a set $R$ with two binary operations, addition and multiplication, such that:
\[\begin{aligned} \text{(R1)}&\quad (R, +, 0, -(-)) \text{ is a commutative group} \\ \text{(R2)}&\quad (a \cdot b) \cdot c = a \cdot (b \cdot c) \quad \text{(multiplication is associative)} \\ \text{(R3)}&\quad \exists\, 1 \in R,\ 1 \cdot a = a \text{ and } a \cdot 1 = a \quad \text{(multiplicative identity)} \\ \text{(R4)}&\quad a \cdot (b + c) = a \cdot b + a \cdot c, \quad (a + b) \cdot c = a \cdot c + b \cdot c \quad \text{(distributivity)} \end{aligned}\]Some textbooks do not require a multiplicative identity. Such a structure is called a rng — the missing “i” stands for the missing identity. This book includes the identity, since that is the more common convention.
Note that (R1) requires commutative addition, unlike the general Group of
Chapter 6. Hence, before defining Ring, we first define what “commutative”
means as an extension of Group.
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