query: Let $R$ be a ring with identity. Prove that if $1-ab$ is invertible for some $a,b \in R$, then $1-ba$ is also invertible.
positive_passages.docid: 146795
positive_passages.text: Proof If $AB-I$ Invertible then $BA-I$ invertible.
negative_passages.docid: 116651
negative_passages.text: Show for any simple ring without identity, R, that R is a division ring.
negative_passages.docid: 127418
negative_passages.text: If ring $B$ is integral over $A$, then an element of $A$ which is a unit in $B$ is also a unit in $A$.
negative_passages.docid: 99631
negative_passages.text: Prove/Disprove: Find $a\in S_4$ that is not a power of $b$ such that $ab=ba$ while $b=(123)$
negative_passages.docid: 67208
negative_passages.text: Can we say that there exist an integer $n$ such $A+nB$ invertible?
negative_passages.docid: 17032
negative_passages.text: If $ab=ba$, Prove $a^2$ commutes with $b^2$
negative_passages.docid: 82925
negative_passages.text: Let the matrix $A=[a_{ij}]_{n×n}$ be defined by $a_{ij}=\gcd(i,j )$. How prove that $A$ is invertible, and compute $\det(A)$?
negative_passages.docid: 64742
negative_passages.text: Prove a residue matrix $A$ (with coefficients in $\mathbb Z_n)$ has an inverse if and only if $\gcd(\det A,n) = 1$
negative_passages.docid: 53113
negative_passages.text: $I-AB$ be invertible $\Leftrightarrow$ $I-BA$ is invertible
negative_passages.docid: 5758
negative_passages.text: Prove that $I+\lambda A$,$I+\lambda B$,$I+\lambda C$ are invertible and A=B=C
negative_passages.docid: 157784
negative_passages.text: Prove that if $\gcd(ab,c)=1$, then $\gcd(a,c)=1$.
negative_passages.docid: 39788
negative_passages.text: $a+1,a-1$ invertible for nilpotent element
negative_passages.docid: 106398
negative_passages.text: If $A$ is invertible show that $\det(A) \neq 0$
negative_passages.docid: 52357
negative_passages.text: If $a,b \in G$, show that $ab^{n}a^{-1}=(aba^{-1})^{n}$.
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: 25488
negative_passages.text: Let $A,B$ be $m\times m$ matrices such that $AB$ is invertible. Show $A,B$ invertible.
negative_passages.docid: 110944
negative_passages.text: Proving that $AB-BA=cI$ for nontrivial $c \in \mathbb{C}$