Unitary R-Modules with n-Element Bases are Isomorphic
Jump to navigation
Jump to search
Theorem
Let $G$ and $H$ be unitary $R$-modules for some ring with unity $R$.
Let $G$ and $G$ both have bases with $n$ elements.
Then $G$ and $H$ are isomorphic.
Proof
From Isomorphism from $R^n$ via $n$-Term Sequence, both $G$ and $H$ are isomorphic to the $R$-module $R^n$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases: Theorem $27.5$: Passim