query: In which rings $R$ does $a\mid xy$ imply the existence of $b\mid x$ and $c\mid y$ with $a=bc$?
positive_passages.docid: 23760
positive_passages.text: In what kind of rings a divisor of a product is a product of divisors?
negative_passages.docid: 105887
negative_passages.text: Ring and Subring with different Identities
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: 6896
negative_passages.text: If $A/I \cong A/J$ as rings and $I\subseteq J,$ then $I=J.$
negative_passages.docid: 24845
negative_passages.text: Let $a\mid c$ and $b\mid c$ such that $\gcd(a,b)=1$, Show that $ab\mid c$
negative_passages.docid: 29019
negative_passages.text: Ideals in direct product of rings
negative_passages.docid: 16945
negative_passages.text: Finding the Ideals of a Direct Product of Rings
negative_passages.docid: 18971
negative_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: 155445
negative_passages.text: What kind of ring $R$ makes $\mathcal O(\mathrm{Proj}\,R[x_0,\cdots,x_n])=R$?
negative_passages.docid: 8787
negative_passages.text: In a ring $(A,+, \cdot)$ if $aba = a$ then $bab = b$ and all non zero elements in $A$ are invertible.
negative_passages.docid: 30583
negative_passages.text: If $a\mid b$ and $b\mid a$, $a,b\in R$, show that $\exists w \in R^\times$ with $b=wa$
negative_passages.docid: 10908
negative_passages.text: Associated elements in a ring
negative_passages.docid: 30714
negative_passages.text: A question on ring homomorphisms and maximal ideals.
negative_passages.docid: 135779
negative_passages.text: Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$.
negative_passages.docid: 87078
negative_passages.text: Prove the following set is an ideal of the ring $Z[x]$
negative_passages.docid: 106894
negative_passages.text: If a subring of a ring R has identity, does R also have the identity?
negative_passages.docid: 56512
negative_passages.text: Divisibility is transitive: $\ a\mid b\mid c\,\Rightarrow\ a\mid c$