query: Prove that if $R$ is a commutative ring, then $R[x]$ is never a field
positive_passages.docid: 46254
positive_passages.text: Why can't the Polynomial Ring be a Field?
negative_passages.docid: 108955
negative_passages.text: Showing $(\mathbb{Q},+)$ is not isomorphic to $(\mathbb{R},+)$
negative_passages.docid: 68807
negative_passages.text: How to show that the ring $S/A$ has no zero divisors? (Hungerford, Algebra, Problem 12, Chapter III, Section 2)
negative_passages.docid: 46843
negative_passages.text: A ring without the Invariant Basis Number property
negative_passages.docid: 79897
negative_passages.text: if R is a commutative ring in which all the prime ideals are finitely generated then R is Noetherian
negative_passages.docid: 70712
negative_passages.text: How to prove that $k[x, xy, xy^2, \dotsc]$ is not noetherian?
negative_passages.docid: 149747
negative_passages.text: Prove that if I is a prime ideal of R then I[x] is a prime ideal of R[x]
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: 26805
negative_passages.text: Non-Noetherian 0-dimensional ring
negative_passages.docid: 13983
negative_passages.text: Show that If $R[X]$ is Euclidean domain then $R$ is a field
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: 155604
negative_passages.text: Show that $k[x,y]/(xy-1)$ is not isomorphic to a polynomial ring in one variable.
negative_passages.docid: 48537
negative_passages.text: If $\forall x \in R, x^2-x \in Z(G)$, than $R$ is commutative
negative_passages.docid: 156457
negative_passages.text: Show that $K[x,xy,xy^2,\dots]$ is not Noetherian
negative_passages.docid: 73248
negative_passages.text: Prove that $\mathbb{R^*}$, the set of all real numbers except $0$, is not a cyclic group
negative_passages.docid: 75315
negative_passages.text: How to prove that homomorphism from field to ring is injective or zero?
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