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Rings, fields, spaces etc.

  1. Jun 21, 2010 #1
    They all seem to be defined as sets with multiplication and addition axioms satisfied. What is the difference?
  2. jcsd
  3. Jun 21, 2010 #2
    A field is a ring where every nonzero element has a multiplicative inverse. All fields are rings, but not vice-versa. What spaces are you talking about, vector spaces?
  4. Jun 21, 2010 #3
    Maybe spaces is not accurate but there seem to be a lot of things which are defined as having satisfying similar axioms.
  5. Jun 21, 2010 #4
    Yes, it's true, but they do all have their differences. Vector spaces, for example, need both a set of vectors and a field of scalars. You can treat a field as a vector space over itself, because of the similarity of the axioms, but they are intrinsically different.
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