Group Ring Field F-vector spaces
https://www.reddit.com/r/math/comments/xmcy3h/why_is_a_vector_space_defined_over_a_field/
Why are vector spaces defined over a field?
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Vectors spaces over other algebraic structures have different properties. For example, if we define it over any Abelian ring, then it may not have a basis. The anthesis to defining a vector space is the guarantee that a basis exists. With the existence of a basis, we are able to prove many fascinating properties. Vector Spaces over any other structure may not guarantee a basis.
- Vector spaces over a ring are known as modules. The problem with modules is that there are some modules without a basis. The use of a field ensures a basis!