Difference between revisions of "Uniqueness of Rank of Free Modules Over Commutative Rings"
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| + | ==Theorem== | ||
Let <math>R</math> be a commutative ring with <math>1 \ne 0</math> and let M be a free R-module. If A and B are both bases for M, then A and B | Let <math>R</math> be a commutative ring with <math>1 \ne 0</math> and let M be a free R-module. If A and B are both bases for M, then A and B | ||
have the same cardinality, meaning that there exists a bijection <math>A → B</math>. | have the same cardinality, meaning that there exists a bijection <math>A → B</math>. | ||
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| + | [[Category: Free Modules]] [[Category: Theorems]] | ||
Latest revision as of 21:50, 9 March 2023
Theorem
Let \(R\) be a commutative ring with \(1 \ne 0\) and let M be a free R-module. If A and B are both bases for M, then A and B have the same cardinality, meaning that there exists a bijection \(A → B\).
