Rings, fields, spaces etc.

1. Jun 21, 2010

Gregg

They all seem to be defined as sets with multiplication and addition axioms satisfied. What is the difference?

2. Jun 21, 2010

Tedjn

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?

3. Jun 21, 2010

Gregg

Maybe spaces is not accurate but there seem to be a lot of things which are defined as having satisfying similar axioms.

4. Jun 21, 2010

Tedjn

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.