# External Direct Sum of Rings is Ring

## Theorem

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

Then their (external) direct product:

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

is a ring.

## Proof

Consider the structures $\struct {R_1, +_1}, \struct {R_2, +_2}, \ldots, \struct {R_n, +_n}$.

By the definition of a ring, these are all groups.

From External Direct Product of Groups is Group we have that the their external direct product:

- $\ds \struct {R, +} = \prod_{k \mathop = 1}^n \struct {R_k, +_k}$

is a group.

Similarly, consider the structures $\struct {R_1, \circ_1}, \struct {R_2, \circ_2}, \ldots, \struct {R_n, \circ_n}$.

By the definition of a ring, these are all semigroups.

From External Direct Product of Semigroups we have that the their external direct product:

- $\ds \struct {R, \circ} = \prod_{k \mathop = 1}^n \struct {R_k, \circ_k}$

is a semigroup.

Finally we note that from External Direct Product of Ringoids is Ringoid, $\circ$ as defined here is distributive over $+$.

Hence the result, by definition of ring.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old: Theorem $22.8$ - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 3.1$: Direct sums