query: Let $A$ be a local noetherian ring and $\mathcal{M}$ is the maximal ideal of $A$. Then $A/\mathcal{M}^n$ is artinian.
positive_passages.docid: 45684
positive_passages.text: Is quotient of a ring by a power of a maximal ideal local?
negative_passages.docid: 49521
negative_passages.text: In an extension of finitely generated $k$-algebras the contraction of a maximal ideal is also maximal
negative_passages.docid: 46717
negative_passages.text: If $M$ is a Noetherian $R$-module, then $R/\text{Ann}(M)$ is a Noetherian ring
negative_passages.docid: 69352
negative_passages.text: Prove that $I= \{a+bi \in ℤ[i] : a≡b \pmod{2}\}$ is an maximal ideal of $ℤ[i]$.
negative_passages.docid: 27766
negative_passages.text: Comaximal ideals in a commutative ring
negative_passages.docid: 43230
negative_passages.text: Is $\langle1-i\rangle$ a maximal ideal in $\mathbb{Z}[i]$?
negative_passages.docid: 114193
negative_passages.text: On the absolute norm of an ideal
negative_passages.docid: 96144
negative_passages.text: Every element outside the maximal ideal of a local ring is a unit
negative_passages.docid: 116883
negative_passages.text: About ideals in a ring which is a $k$ vector space
negative_passages.docid: 129025
negative_passages.text: Quotient ring local if Ring is local
negative_passages.docid: 133975
negative_passages.text: Localising a polynomial ring and non-maximal prime ideal
negative_passages.docid: 131497
negative_passages.text: Maximal ideal not containing the set of powers of an element is prime
negative_passages.docid: 121750
negative_passages.text: Let $R$ be a commutative, unital ring and let $I$ be an ideal of $R.$ Prove that if $I$ is maximal then $R/I$ is a field.
negative_passages.docid: 6699
negative_passages.text: An ideal that is maximal among non-finitely generated ideals is prime.
negative_passages.docid: 131943
negative_passages.text: How to directly prove that $M$ is maximal ideal of $A$ iff $A/M$ is a field?
negative_passages.docid: 68202
negative_passages.text: Must an ideal generated by an irreducible element be a maximal ideal?
negative_passages.docid: 109199
negative_passages.text: Prove that every nonzero prime ideal is maximal in $\mathbb{Z}[\sqrt{d}]$