Is vector space a field? And what are: Groups / Rings / Fields / Vector Spaces?

Quote from Lecture 3, 6.S897 Algebra and Computation by Madhu Sudan, MIT:

Suppose we have a vector space $(V,+,\circ)$ where:

• $V$ is a set of vectors
• $+$ is vector addition operator: $V \times V \mapsto V$
  • E.g. $\vec a + \vec b = \vec c$
• $\circ$ is vector scaling operator: $F \times V \mapsto V$
  • E.g. $c \circ \vec v = \vec u$

Note that, dot product $\cdot$ and cross product $\times$ are NOT part of the vector space definition!

We don’t have to follow the notation in the note above, so, taking the definition of Rings, we have

• Case 1: $(V, +)$ corresponds to $(R, +)$ and $(V, \circ)$ to $(R, \cdot)$
  • OR
• Case 2: $(V, +)$ corresponds to $(R, \cdot)$ and $(V, \circ)$ to $(R, +)$

Case 1:

• $(V, +)$ is an Abelian group. (satisfies condition 1)
• $(V, \circ)$ is NOT a monoid. (fails condition 2)

Case 2:

• $(V, \circ)$ is NOT a monoid. (fails condition 1)

So a vector space is not even a ring. Of course it cannot be a field.

A side dish of this post is that we now know $(V, +)$ is an Abelian group.