Examples
- If we think of as the set of real numbers, then the direct product is precisely just the cartesian product, .
- If we think of as the group of real numbers under addition, then the direct product still consists of . The difference between this and the preceding example is that is now a group. We have to also say how to add their elements. This is done by letting .
- If we think of as the ring of real numbers, then the direct product again consists of . To make this a ring, we say how their elements are added, and how they are multiplied .
- However, if we think of as the field of real numbers, then the direct product does not exist - naively defining in a similar manner to the above examples would not result in a field since the element does not have a multiplicative inverse.
In a similar manner, we can talk about the product of more than two objects, e.g. . We can even talk about product of infinitely many objects, e.g. .
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