**Note:** Some authors use the term **module category** for the category of modules; this term can be ambiguous since it could also refer to a category with a monoidal-category action.^{}

The category of left modules (or that of right modules) is an abelian category. The category has enough projectives^{} and enough injectives.^{} Mitchell's embedding theorem states every abelian category arises as a full subcategory of the category of modules.

Projective limits and inductive limits exist in the category of (say left) modules.^{}

Over a commutative ring, together with the tensor product of modules ⊗, the category of modules is a symmetric monoidal category.

The category **K-Vect** (some authors use **Vect**_{K}) has all vector spaces over a fixed field *K* as objects and *K*-linear transformations as morphisms. Since vector spaces over *K* (as a field) are the same thing as modules over the ring *K*, **K-Vect** is a special case of **R-Mod**, the category of left *R*-modules.

This page was last edited on 3 March 2018, at 07:38 (UTC).

Reference: https://en.wikipedia.org/wiki/Category_of_finite_dimensional_K-vector_spaces under CC BY-SA license.

Reference: https://en.wikipedia.org/wiki/Category_of_finite_dimensional_K-vector_spaces under CC BY-SA license.

- Ring
- Modules
- Module Homomorphisms
- Category Of Abelian Groups
- Module Category
- Monoidal-category Action
- Abelian Category
- Enough Projectives
- Enough Injectives
- Mitchell's Embedding Theorem
- Projective Limits
- Inductive Limits
- Tensor Product Of Modules
- Symmetric Monoidal Category
- Category
- Vector Spaces
- Field
- Objects
- K-linear Transformations
- Morphisms
- Modules
- Ring

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