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

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

Sources

• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 3.1$: Direct sums: Definition $3.1$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.3$: Congruences