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Definition Punctuation

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Indigo5684
2025-09-30 13:19:35 -05:00
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docs/math/abstract-algebra/DF-10-modules.md
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@@ -15,7 +15,7 @@
**Remark**. Modules over a field $F$ and vector spaces over $F$ are identical.
**Definition** An *R-submodule* is a subset$N \subseteq M$ which is closed under the action taken forall $r \in R$. That is, given $r \in R, n \in N$, then $rn \in N$. Every module has at least two submodules: itself and the trivial (empty) submodule.
**Definition**. An *R-submodule* is a subset$N \subseteq M$ which is closed under the action taken forall $r \in R$. That is, given $r \in R, n \in N$, then $rn \in N$. Every module has at least two submodules: itself and the trivial (empty) submodule.
**Remark**. If $F$ is a field, submodules are equivalent to subspaces.
@@ -124,7 +124,7 @@ This direct product is in itself an $R$-module.
**Corollary**. If $F$ is a free $R$-module with basis $A$, then $F \cong F(A)$.
**Definition** For a free module $F$ with basis $A$, if $R$ is commutative, then the *rank* of $F$ is the cardinality of $A$.
**Definition**. For a free module $F$ with basis $A$, if $R$ is commutative, then the *rank* of $F$ is the cardinality of $A$.
## Section 10.4 - Tensor Products of Modules