query: Let $R$ be a ring with the property that whenever $a, b, c ∈ R$ with a $a\neq 0$ and $ab = ca$, it follows that $b = c$. Is $R$ a domain?
positive_passages.docid: 18971
positive_passages.text: Show that a ring is commutative if it has the property that ab = ca implies b = c when $a\neq 0$
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: 66451
negative_passages.text: In a ring, how do we prove that a * 0 = 0?
negative_passages.docid: 78626
negative_passages.text: Let $R$ be a commutative ring. If $R[X]$ is a principal ideal domain, then $R$ is a field.
negative_passages.docid: 21604
negative_passages.text: A complete ring with respect to an ideal
negative_passages.docid: 154495
negative_passages.text: Prove that a nontrivial subring of a ring that is a domain has the same identity
negative_passages.docid: 5594
negative_passages.text: Question about the Euclidean ring definition
negative_passages.docid: 115920
negative_passages.text: Showing $R$ is a local ring if and only if all elements of $R$ that are not units form an ideal
negative_passages.docid: 84189
negative_passages.text: Is $\mathbb{C}$ equal to $\mathbb{R}^2$?
negative_passages.docid: 138337
negative_passages.text: On a ring $R$ where $x^2 = x$, we must have that $2x = 0$ and $R$ is commutative
negative_passages.docid: 14158
negative_passages.text: Prove that $R$ is an integral domain $\Leftrightarrow$ $R[x]$ is an integral domain
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: 116453
negative_passages.text: Is any quotient of a Euclidean domain by a prime ideal a Euclidean domain?
negative_passages.docid: 3936
negative_passages.text: Characterization of valuation rings
negative_passages.docid: 13983
negative_passages.text: Show that If $R[X]$ is Euclidean domain then $R$ is a field
negative_passages.docid: 96132
negative_passages.text: $R$ integral domain : $u\in R^*, a \text{ is prime} \iff au \text{ is prime}$
negative_passages.docid: 133495
negative_passages.text: Example of a ring without Invariant Basis Property