What Distinguishes Rings, Fields, and Spaces in Mathematics?

[Gregg]

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

 

[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?

 

[Gregg]

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

 

[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.