Definition:Ring Direct Product/Finite Case

Definition
Let $\struct {R_1, +_1, \circ_1}, \struct {R_2, +_2, \circ_2}, \ldots, \struct {R_n, +_n, \circ_n}$ be rings.

Let:
 * $\ds \struct {R, +, \circ} = \prod_{k \mathop = 1}^n \struct {R_k, +_k, \circ_k}$

be the external direct product on two operations, such that:


 * $(1):\quad$ The operation $+$ induced on $R$ by $+_1, \ldots, +_n$ is defined as:
 * $\tuple {s_1, s_2, \ldots, s_n} + \tuple {t_1, t_2, \ldots, t_n} = \tuple {s_1 +_1 t_1, s_2 +_2 t_2, \ldots, s_n +_n t_n}$


 * $(2):\quad$ The operation $\circ $ induced on $R$ by $\circ_1, \ldots, \circ_n$ is defined as:
 * $\tuple {s_1, s_2, \ldots, s_n} \circ \tuple {t_1, t_2, \ldots, t_n} = \tuple {s_1 \circ_1 t_1, s_2 \circ_2 t_2, \ldots, s_n \circ_n t_n}$

Then $\struct {R, +, \circ}$ is referred to as the (external) direct product of $\struct {R_1, +_1, \circ_1}, \struct {R_2, +_2, \circ_2}, \ldots, \struct {R_n, +_n, \circ_n}$.

The external direct product of $R_1, R_2, \ldots, R_n$ is often denoted
 * $R_1 \times R_2 \times \cdots \times R_n$