F-vector space in Abstract Algebra
Definition
A $F$-vector space is a $R$-module that is a field in the context of a ring $R = F$.
Explanation
This is a definition that even feels sophisticated, considering that a module is a generalization of a vector field, it makes sense.
See Also
The $F$-vector spaces discussed in the documents below are essentially no different from the vector spaces mentioned in the above documents. However, the perspective is somewhat different; vector spaces in linear algebra are an abstraction of the intuitive Euclidean space, while vector spaces in abstract algebra bring it into the realm of ‘algebra’ in the true sense.
Conversely, a $R$-module generalizes the scalar field of a $F$-vector space to a scalar ring $R$, and thus its naming reflects its identity without regard to the history and meaning of $F$-vector spaces. From the perspective of a group $G$, it’s like adding a ring $R$ and a new operation $\mu$, hence it’s also called an additive group.