query: Non-principal ideal in ring $\Bbb Z×\Bbb Z$
positive_passages.docid: 149783
positive_passages.text: $\mathbb Z\times\mathbb Z$ is principal but is not a PID
negative_passages.docid: 155745
negative_passages.text: Ideals of $\mathbb{Z}[X]$
negative_passages.docid: 20296
negative_passages.text: The ideal $I= \langle x,y \rangle\subset k[x,y]$ is not principal
negative_passages.docid: 53367
negative_passages.text: In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal. What could be wrong in this approach?
negative_passages.docid: 99198
negative_passages.text: Principal ideal and free module
negative_passages.docid: 24351
negative_passages.text: The ring $\Bbb Z\left [\frac{-1+\sqrt{-19}}{2}\right ]$ is not a Euclidean domain
negative_passages.docid: 9794
negative_passages.text: Quotient ring of Gaussian integers $\mathbb{Z}[i]/(a+bi)$ when $a$ and $b$ are NOT coprime
negative_passages.docid: 70047
negative_passages.text: Is the ideal generated by irreducible element in principal ideal domain maximal?
negative_passages.docid: 111307
negative_passages.text: If ideal $I$ of domain $R$ is free $R$-module, then $I$ is principal ideal.
negative_passages.docid: 65536
negative_passages.text: Finitely generated ideals in a Boolean ring are principal, why?
negative_passages.docid: 1719
negative_passages.text: Ideals in the ring of Gaussian integers
negative_passages.docid: 133975
negative_passages.text: Localising a polynomial ring and non-maximal prime ideal
negative_passages.docid: 66169
negative_passages.text: Ring of order $n$ is isomorphic to $\Bbb Z/n\Bbb Z$, with $n$ square-free
negative_passages.docid: 102341
negative_passages.text: Primary decomposition of a monomial ideal
negative_passages.docid: 127030
negative_passages.text: Simple example of non-arithmetic ring (non-distributive ideal lattice)
negative_passages.docid: 87151
negative_passages.text: Primary ideals in Noetherian rings
negative_passages.docid: 17075
negative_passages.text: Show an Ideal is the principal ideal for some polynomial.