query: The number of elements in a linear independent set and the basis with a free module over a commutative ring with identity.
positive_passages.docid: 28136
positive_passages.text: $A^m\hookrightarrow A^n$ implies $m\leq n$ for a ring $A\neq 0$
negative_passages.docid: 6646
negative_passages.text: A (non-artificial) example of a ring without maximal ideals
negative_passages.docid: 118781
negative_passages.text: product of ideals in matrix ring
negative_passages.docid: 73743
negative_passages.text: Number of elements in the quotient ring $\mathbb{Z}[X]/(X^2-3, 2X+4)$
negative_passages.docid: 80616
negative_passages.text: A ring with few invertible elements
negative_passages.docid: 10908
negative_passages.text: Associated elements in a ring
negative_passages.docid: 87078
negative_passages.text: Prove the following set is an ideal of the ring $Z[x]$
negative_passages.docid: 73825
negative_passages.text: Every $R$-module is a quotient of a free module
negative_passages.docid: 28402
negative_passages.text: Number of bases of an n-dimensional vector space over q-element field.
negative_passages.docid: 15000
negative_passages.text: Show that number of elements in $(Z/nZ)^*$ is $\phi (n)$
negative_passages.docid: 7303
negative_passages.text: How many idempotent elements does the ring ${\bf Z}_n$ contain?
negative_passages.docid: 102721
negative_passages.text: Number of elements in the ring $\mathbb Z [i]/\langle 2+2i\rangle$
negative_passages.docid: 21604
negative_passages.text: A complete ring with respect to an ideal
negative_passages.docid: 49854
negative_passages.text: Ideals in polynomial rings generated by linear polynomials
negative_passages.docid: 86922
negative_passages.text: Ring with 10 elements is isomorphic to $\mathbb{Z}/10 \mathbb{Z} $
negative_passages.docid: 45576
negative_passages.text: A surjective homomorphism between finite free modules of the same rank
negative_passages.docid: 83116
negative_passages.text: Uncountability of basis of $\mathbb R^{\mathbb N}$