query: Is there a ring which satisfies $xy=1$ and $yx\neq 1$
positive_passages.docid: 17059
positive_passages.text: A ring element with a left inverse but no right inverse?
negative_passages.docid: 124456
negative_passages.text: Given $x \in \mathbb{R}$ and $N \in \mathbb{N}, N>1,$ find integers $h,k$ with $0 < k \leq N $ that satisfy $\lvert{kx - h\rvert} < 1/N.$
negative_passages.docid: 35066
negative_passages.text: Nature and number of solutions to $xy=x+y$
negative_passages.docid: 148804
negative_passages.text: Norm-Euclidean rings?
negative_passages.docid: 35777
negative_passages.text: $\mathbb{R}[X,Y]/(Y-X^2,Y+X)\cong_\text{Ring} \mathbb{R}\times \mathbb{R}$
negative_passages.docid: 6121
negative_passages.text: Quotient ring of Gaussian integers
negative_passages.docid: 21538
negative_passages.text: Can a ring of positive characteristic have infinite number of elements?
negative_passages.docid: 512
negative_passages.text: Finding generator for the ideal generated by $a$ and $b$ in an euclidean domain
negative_passages.docid: 140190
negative_passages.text: Binary operation $xy$, has identity, but not associativity. Is the inverse unique?
negative_passages.docid: 152643
negative_passages.text: $(xy + 1)(yz + 1)(zx + 1) $ is a perfect square if and only if each factor is
negative_passages.docid: 34668
negative_passages.text: Find all rings $R$ satisfying $\mathbb Z \subset R \subset \mathbb Z[\frac {1}{2}]$
negative_passages.docid: 120213
negative_passages.text: Example $x$, $y$ and $z$ values for $x\uparrow^\alpha y=z$ where $\alpha\in \Bbb R-\Bbb N$
negative_passages.docid: 129438
negative_passages.text: Gaussian Integers and Quotient Rings
negative_passages.docid: 133495
negative_passages.text: Example of a ring without Invariant Basis Property
negative_passages.docid: 32741
negative_passages.text: Idempotents in a ring without unity (rng) and no zero divisors.
negative_passages.docid: 130073
negative_passages.text: $N(ab) \geq N(a)$ is not necessary for the ring to be euclidean
negative_passages.docid: 46047
negative_passages.text: Check whether the quotient ring $R:=\mathbb{C}[x,y,z]/(xy-z^2)$ is a field or an integral domain