Direct Product - Examples

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. .

Read more about this topic:  Direct Product

Famous quotes containing the word examples:

    “Histories are more full of examples of the fidelity of dogs than of friends.”
    —Alexander Pope (1688–1744)

    “No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
    —André Breton (1896–1966)

    “In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
    —Michel de Montaigne (1533–1592)