query: On $\mathbb{Z}[x]$ polynomial rings is module $M=(x,3)=x\mathbb{Z}[x]+3\mathbb{Z}[x]$ cyclic?
positive_passages.docid: 1720
positive_passages.text: Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z [x]$
negative_passages.docid: 52212
negative_passages.text: If $\alpha$ is an algebraic element and $L$ a field, does the polynomial ring $L[\alpha]$ is also a field?
negative_passages.docid: 58532
negative_passages.text: If $M$ is an Artinian $R$-module then $R/\mbox{ann}(M)$ is an Artinian ring?
negative_passages.docid: 142470
negative_passages.text: $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})= \mathbb{Q}(\sqrt{p}\cdot \sqrt[3]{q})$ ??
negative_passages.docid: 117970
negative_passages.text: Find the minimal polynomial for $\sqrt[3]{2} + \sqrt[3]{4}$ over $\mathbb{Q}$
negative_passages.docid: 20097
negative_passages.text: Cardinality of the quotient ring $\mathbb{Z}[x]/(x^2-3,2x+4)$
negative_passages.docid: 65637
negative_passages.text: Squares in Polynomial Quotient Ring $ (\mathbb{Z}/2)[x] /(x^{17} - 1) $
negative_passages.docid: 95146
negative_passages.text: Polynomial ring $R[X] $ has zero Jacobson radical, for $R$ a domain
negative_passages.docid: 25290
negative_passages.text: Determine the number of irreducible monic polynomials of degree 3 in $\mathbb F_p[x]$
negative_passages.docid: 109691
negative_passages.text: When are kernels of homomorphisms on $\mathbb{Z}[x]$ principal ideals, and what does this have to do with Gauss's lemma?
negative_passages.docid: 136804
negative_passages.text: Computing $\mathrm{Hom}(\mathbb Z_n,\mathbb Z_m)$ as $\mathbb Z$-module
negative_passages.docid: 37929
negative_passages.text: Classification of prime ideals of $\mathbb{Z}[X]$
negative_passages.docid: 14003
negative_passages.text: $(\mathbb{Z}/2^n \mathbb{Z})^*$ is not cyclic Group for $n\geq 3$
negative_passages.docid: 19112
negative_passages.text: Ring $R= \mathbb Z_3[x]/\langle x^6-1\rangle$
negative_passages.docid: 14191
negative_passages.text: How many elements does the ring $ℤ[X]/(X^2-3,2X-4)$ have? Describe the structure of this ring.
negative_passages.docid: 60391
negative_passages.text: Ideal Generated by Three Elements in Polynomial Ring
negative_passages.docid: 77862
negative_passages.text: Question about fields and quotients of polynomial rings