query: Is it true that $R^n\simeq R^m$ as rings implies $m=n$?
positive_passages.docid: 22604
positive_passages.text: Can $R \times R$ be isomorphic to $R$ as rings?
negative_passages.docid: 63338
negative_passages.text: Noncommutative failure of IBN: $\,R^n\cong R^m \Rightarrow n - m,$ ring $R$ with maximal ideal $I$ such that $R/I$ is not a division ring?
negative_passages.docid: 137739
negative_passages.text: Prove that the group isomorphism $\mathbb{Z}^m \cong \mathbb{Z}^n$ implies that $m = n$
negative_passages.docid: 118073
negative_passages.text: Is the ring $m\mathbb{Z}$ isomorphic to the ring $n\mathbb{Z}$?
negative_passages.docid: 82953
negative_passages.text: In a ring with no zero-divisors, for $(m,n) =1$, $a^m = b^m$ and $a^n = b^n$ $\iff a =b$
negative_passages.docid: 75520
negative_passages.text: If $a^3 =a$ for all $a$ in a ring $R$, then $R$ is commutative.
negative_passages.docid: 70576
negative_passages.text: Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$
negative_passages.docid: 109532
negative_passages.text: Proving $5^n \equiv 1 \pmod {2^r}$ when $n=2^{r-2}$
negative_passages.docid: 155445
negative_passages.text: What kind of ring $R$ makes $\mathcal O(\mathrm{Proj}\,R[x_0,\cdots,x_n])=R$?
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: 122622
negative_passages.text: Does there exist a matrix $P$ such that $P^n=M$ for a special matrix $M$?
negative_passages.docid: 49166
negative_passages.text: $R^n \cong R^m$ iff $n=m$
negative_passages.docid: 18330
negative_passages.text: Does $R[x] \cong S[x]$ imply $R \cong S$?
negative_passages.docid: 31331
negative_passages.text: $|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power
negative_passages.docid: 46355
negative_passages.text: If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
negative_passages.docid: 5893
negative_passages.text: On the commutative property of multiplication (domain of integers, possibly reals)
negative_passages.docid: 41398
negative_passages.text: For a ring $R$, does $\operatorname{End}_R(R)\cong R^{\mathrm{op}}$?