Definition:Ring Direct Product

Definition

Finite Case

Let $\left({R_1, +_1, \circ_1}\right), \left({R_2, +_2, \circ_2}\right), \ldots, \left({R_n, +_n, \circ_n}\right)$ be rings.

Let:

$\displaystyle \left({R, +, \circ}\right) = \prod_{k \mathop = 1}^n \left({R_k, +_k, \circ_k}\right)$

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

$(1):\quad$ The operation $+$ induced on $R$ by $+_1, \ldots, +_n$ is defined as:

$\left({s_1, s_2, \ldots, s_n}\right) + \left({t_1, t_2, \ldots, t_n}\right) = \left({s_1 +_1 t_1, s_2 +_2 t_2, \ldots, s_n +_n t_n}\right)$

$(2):\quad$ The operation $\circ $ induced on $R$ by $\circ_1, \ldots, \circ_n$ is defined as:

$\left({s_1, s_2, \ldots, s_n}\right) \circ \left({t_1, t_2, \ldots, t_n}\right) = \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2, \ldots, s_n \circ_n t_n}\right)$

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

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

$R_1 \times R_2 \times \cdots \times R_n$