query: What are some examples of free modules that have 2 bases of different sizes?
positive_passages.docid: 25764
positive_passages.text: Example of a ring such that $R^2\simeq R^3$, but $R\not\simeq R^2$ (as $R$-modules)
negative_passages.docid: 17978
negative_passages.text: Isomorphism classes of $\mathbb{Z}[i]$ modules.
negative_passages.docid: 29756
negative_passages.text: Classsifying 1- and 2- dimensional Algebras, up to Isomorphism
negative_passages.docid: 76501
negative_passages.text: How can we show that $\mathbb Q$ is not a free $\mathbb Z$-module?
negative_passages.docid: 120637
negative_passages.text: Is there a unique cyclic submodule for every factor dividing the order of a cyclic module?
negative_passages.docid: 7010
negative_passages.text: Example of two non-isomorphic fields which embed inside each other
negative_passages.docid: 139861
negative_passages.text: Support of a module with extended scalars
negative_passages.docid: 23818
negative_passages.text: Divisibility Rules for Bases other than $10$
negative_passages.docid: 154142
negative_passages.text: Finding a specific basis for an endomorphism
negative_passages.docid: 48519
negative_passages.text: Symmetric powers of modules
negative_passages.docid: 87750
negative_passages.text: Hom / tensor adjunction for $O_X$ modules?
negative_passages.docid: 18003
negative_passages.text: Standard example where Jacobson radical not equal nilradical
negative_passages.docid: 95942
negative_passages.text: $k[x]/(x^n)$ module with finite free resolution is free
negative_passages.docid: 86724
negative_passages.text: Example of a non-zero module that has no associated primes
negative_passages.docid: 74974
negative_passages.text: When $\operatorname{Hom}_{R}(M,N)$ is finitely generated as $\mathbb Z$-module or $R$-module?
negative_passages.docid: 103544
negative_passages.text: Determine a formula for a dual basis.
negative_passages.docid: 124864
negative_passages.text: A not free $\mathbb{Z}$-module