query: for every $x$ in ring $R$, we have $x^2 = x$. Show any finitely generated ideal is cyclic.
positive_passages.docid: 149519
positive_passages.text: Prove that a Boolean ring is a Bezout ring .
negative_passages.docid: 140553
negative_passages.text: An ideal which is not finitely generated
negative_passages.docid: 2104
negative_passages.text: Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic.
negative_passages.docid: 35664
negative_passages.text: If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic
negative_passages.docid: 21604
negative_passages.text: A complete ring with respect to an ideal
negative_passages.docid: 89313
negative_passages.text: Maximal ideals of polynomial rings in infinitely many variables
negative_passages.docid: 69728
negative_passages.text: If $f$ divides $g$ in $S[x]$, show that $f$ divides $g$ in $R[x]$ for $R$ a sub-ring of $S$.
negative_passages.docid: 131671
negative_passages.text: Let R be a ring such that $a^2 = a$ , $\forall a$ $\in R$ . Prove that R is commutative.
negative_passages.docid: 69373
negative_passages.text: Prove that $(0)$ is a radical ideal in $\mathbb{Z}/n\mathbb{Z}$ iff $n$ is square free
negative_passages.docid: 130466
negative_passages.text: Proving every prime ideal is maximal in a finite dimensional ring over a field.
negative_passages.docid: 90239
negative_passages.text: $\mathbb{Z} \times \mathbb{Z}$ is cyclic.
negative_passages.docid: 59687
negative_passages.text: Suppose $M$ is a finitely generated module over $R$ where $R$ is a Noetherian ring. Prove that $M$ is a noetherian module.
negative_passages.docid: 149412
negative_passages.text: Every prime ideal of a finitely generated $\mathbb{R}$-algebra is an intersection of maximal ideals?
negative_passages.docid: 95031
negative_passages.text: Find generators of the ideal of an algebraic set
negative_passages.docid: 65536
negative_passages.text: Finitely generated ideals in a Boolean ring are principal, why?
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: 51428
negative_passages.text: Do these two observations suffice to show that a finite boolean ring must be of the form $\mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$?