query: $R$ a $PID \Rightarrow R[[X]]$ a $UFD$
positive_passages.docid: 145351
positive_passages.text: Ring of formal power series over a principal ideal domain is a unique factorisation domain
negative_passages.docid: 19838
negative_passages.text: Prove that in general the set of the right zero divisors is different from the set of left zero divisors in a ring R
negative_passages.docid: 37953
negative_passages.text: Primary Decomposition of $\mathbb{R^n}$ under T
negative_passages.docid: 48792
negative_passages.text: Prove $p_n(x) \rightarrow \sqrt{x}$ uniformly as $n \rightarrow \infty$
negative_passages.docid: 107842
negative_passages.text: Behaviour of ($\overline{a}_n)_{n=1}^{\infty}$ and ($\underline{a}_n)_{n=1}^{\infty}$
negative_passages.docid: 118049
negative_passages.text: Show that $\mathbb{R}^{\mathbb{R}} = U_{e} \oplus U_{o}$
negative_passages.docid: 147546
negative_passages.text: $f: \mathbf{R} \rightarrow \mathbf{R}$ monotone increasing $\Rightarrow$ $f$ is measurable
negative_passages.docid: 138851
negative_passages.text: Euclid's Lemma in a PID: irreducibles are prime: $ \pi\mid ab\Rightarrow \pi\mid a\,$ or $\pi\mid b$
negative_passages.docid: 108060
negative_passages.text: Suppose $f:[0,\infty]\rightarrow$ to R and has continuous $f'(x)$ and $f''(x)$ f(x) → 0 as x → ∞. Show that f′(x)→0 as x→∞
negative_passages.docid: 133775
negative_passages.text: Is $\mathbb{Z}[\sqrt{-7}]$ a UFD or not?
negative_passages.docid: 57138
negative_passages.text: Localization of a UFD that is a PID
negative_passages.docid: 149907
negative_passages.text: $T\circ T=0:V\rightarrow V \implies R(T) \subset N(T)$
negative_passages.docid: 153664
negative_passages.text: When is $\Bbb{Z[\zeta_n]}$ a PID?
negative_passages.docid: 147627
negative_passages.text: Relation between condition numbers $\kappa(A^T A)$ and $\kappa(A)$
negative_passages.docid: 136855
negative_passages.text: Proof that $\mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]$ is a PID
negative_passages.docid: 14158
negative_passages.text: Prove that $R$ is an integral domain $\Leftrightarrow$ $R[x]$ is an integral domain
negative_passages.docid: 30455
negative_passages.text: Show that any continuous $f:[0,1] \rightarrow [0,1]$ has a fixed point $\zeta$