query: Prove that any element in a ring A can be written as a product of irreducible but A is not a UFD
positive_passages.docid: 66238
positive_passages.text: Proving $\mathbb{Z}[\sqrt {10}]$ is not a UFD
negative_passages.docid: 134989
negative_passages.text: Prove that if the identity is written as the product of $r$ transpositions, then $r$ is an even number
negative_passages.docid: 87382
negative_passages.text: Non-invertible elements form an ideal
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: 48009
negative_passages.text: Left inverse implies right inverse in a finite ring
negative_passages.docid: 158343
negative_passages.text: how to prove an element is non-zero in a tensor-product
negative_passages.docid: 28658
negative_passages.text: $R$ with an upper bound for degrees of irreducibles in $R[x]$
negative_passages.docid: 23077
negative_passages.text: Prove that the dihedral group $D_4$ can not be written as a direct product of two groups
negative_passages.docid: 16643
negative_passages.text: A slick proof that a field which is finitely generated as a ring is finite
negative_passages.docid: 92420
negative_passages.text: Characteristic of  a Non-unital Integral Ring
negative_passages.docid: 4240
negative_passages.text: The proof of Triple Vector Products
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: 107821
negative_passages.text: The ring of integers of $\mathbf{Q}[i]$
negative_passages.docid: 42528
negative_passages.text: Prove that every idempotent element is not nilpotent element.
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: 83798
negative_passages.text: Does $A$ a UFD imply that  $A[T]$ is also a UFD?
negative_passages.docid: 6646
negative_passages.text: A (non-artificial) example of a ring without maximal ideals