query: Suppose $R$ is commutative, and let $M$ be a free $R$-module of finite rank. Prove that if $A$ and $B$ are bases of $M$, then $|A| = |B|$.
positive_passages.docid: 133495
positive_passages.text: Example of a ring without Invariant Basis Property
negative_passages.docid: 12474
negative_passages.text: Does an injective endomorphism of a finitely-generated free R-module have nonzero determinant?
negative_passages.docid: 70872
negative_passages.text: Isomorphism of quotient modules implies isomorphism of submodules?
negative_passages.docid: 39652
negative_passages.text: Quotient of free module
negative_passages.docid: 89702
negative_passages.text: Let $R$ be a ring such that for all $a,b$ in $R$, $(a^2-a)b=b(a^2-a)$. Then $R$ is commutative
negative_passages.docid: 133087
negative_passages.text: Prove that, if $\{u,v,w\}$ is a basis for a vector space $V$, then so is $\{u+v, v+w, u+v+w\}$.
negative_passages.docid: 150986
negative_passages.text: For a Noetherian ring $R$, we have $\text{Ass}_R(S^{-1}M)=\text{Ass}_R(M)\cap \{P:P\cap S=\emptyset\}$
negative_passages.docid: 100303
negative_passages.text: A necessary and sufficient condition for $A\cap(B+C)=(A\cap B)+C$ to hold for submodules
negative_passages.docid: 74559
negative_passages.text: $A$ and $B$ are ideals of a ring ${R}$ such that $A\cap B=\{0\}$. Prove that $st=0$ for every $s\in A, t\in B$.
negative_passages.docid: 48537
negative_passages.text: If $\forall x \in R, x^2-x \in Z(G)$, than $R$ is commutative
negative_passages.docid: 37393
negative_passages.text: $R$ is commutative, $I$,$J$ are ideals, $I+J=R$, then $IJ=I\cap J$
negative_passages.docid: 124864
negative_passages.text: A not free $\mathbb{Z}$-module
negative_passages.docid: 66968
negative_passages.text: if $a,b$ are nilpotent elements of a commutative ring $R$, show that $a+b$ is also nilpotent
negative_passages.docid: 4295
negative_passages.text: Let $N$, $K$ be sub-modules of $M$ with $I=\mathrm{Ann}(N)$, $J=\mathrm{Ann}(K)$. Show $I+J$ is a proper subset of $\mathrm{Ann}(N \cap K)$.
negative_passages.docid: 157585
negative_passages.text: Structure theorem of finite rings
negative_passages.docid: 151803
negative_passages.text: Prove that $R$ is commutative
negative_passages.docid: 65224
negative_passages.text: Prove that if $a^2 = a$ for all elements of the ring $R$, then $R$ has characteristic 2. Is the converse of the statement true?