query: $N(ab) \geq N(a)$ is not necessary for the ring to be euclidean
positive_passages.docid: 5594
positive_passages.text: Question about the Euclidean ring definition
negative_passages.docid: 29608
negative_passages.text: If GCD $(a_1,\ldots, a_n)=1$ then there's a matrix in $SL_n(\mathbb{Z})$ with first row $(a_1,\ldots, a_n)$
negative_passages.docid: 46843
negative_passages.text: A ring without the Invariant Basis Number property
negative_passages.docid: 62208
negative_passages.text: If $A$ is a non-empty set, then $A \notin A$
negative_passages.docid: 148804
negative_passages.text: Norm-Euclidean rings?
negative_passages.docid: 55752
negative_passages.text: Prove that $n+1$ vectors in $\mathbb{R}^n$ cannot be linearly independent
negative_passages.docid: 74330
negative_passages.text: $\mathbb{R}^n$ not homeomorphic to open subset of $\mathbb{R}^m$
negative_passages.docid: 73689
negative_passages.text: A question about the equivalence relation on the localization of a ring.
negative_passages.docid: 144572
negative_passages.text: Proof that ideals in $C[0,1]$ are of the form $M_c$ that should not involve Zorn's Lemma
negative_passages.docid: 25682
negative_passages.text: If $A$ is an $m\times n$ matrix, $B$ is an $n\times m$ matrix and $n<m$, then $AB$ is not invertible.
negative_passages.docid: 99782
negative_passages.text: $a_{n+1}=|a_n|-a_{n-1} \implies a_n \; \text{is periodic}$
negative_passages.docid: 32772
negative_passages.text: Proving units in a ring
negative_passages.docid: 131497
negative_passages.text: Maximal ideal not containing the set of powers of an element is prime
negative_passages.docid: 17895
negative_passages.text: Let $R$ be a commutative ring with 1 then why does $a\in N(R) \Rightarrow 1+a\in U(R)$?
negative_passages.docid: 68633
negative_passages.text: Example of Noetherian ring over which the Euclidean algorithm is not valid.
negative_passages.docid: 70576
negative_passages.text: Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$
negative_passages.docid: 65379
negative_passages.text: Deriving $\neg R$ from $\{R↔(R∨(P∧¬P)), R↔¬P, ¬P→(P↔(Q→Q)), P→Q\}$