# Questions tagged [local-rings]

In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.

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### If a finite sum is a unit, then it has a term that is a unit.

Source: Theorem 19.1 (A First Course in Noncommutative Rings by T.Y. Lam) Local Ring on Wikipedia Theorem 19.1 For any nonzero ring R, the following statements are equivalent: (1) $R$ has a unique ...
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### Not-necessarily-unital von Neumann regular local commutative rings: are they fields?

In this question all rings are commutative, but don't necessarily have a multiplicative identity (so: commutative rngs). On Wikipedia there is the unsourced claim: It is well known that a local ring ...
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### If Quotient ring is complete local is the ring complete local?

Let R be a Noetherian commutative ring and N the nil radical. Given R/N is a complete local ring, is R also a complete local ring? If R had two maximal ideal m and n then m+N and n+N are maximal ...
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### Finite dimensional local rings with infinitely many minimal prime ideals

Is there a finite dimensional local ring with infinitely many minimal prime ideals? Equivalent formulation: Is there a ring with a prime ideal $\mathfrak p$ of finite height such that the set of ...
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### local versus graded free resolutions

I'm currently trying to learn about syzygies. Most material is written in the context of graded rings and/or graded modules but I'm interested in a specific question about local rings. Hence I need to ...
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### Is $\mathbb{Q} [X,Y]/[x^{20},y^{20}]$ a local ring?

Is $\mathbb{Q}[X,Y]/[x^{20},y^{20}]$ is a local ring? My approach is to look for a maximal ideal, but got stuck how to find a single maximal ideal. Any feedback would be appreciated.
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### $A/m_a \simeq B/m_b$? Where $A,B$ are local rings, $A\subset B$ and $m_a\subset m_b$

I'm currently reading a math book in french so I'm translating everything as I go and also proving the remarks made throughout. One remark that I haven't been able to prove is: if $A$ and $B$ are ...
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### On the prime spectrum of completion of local rings

Let $(R, \mathfrak m)$ be the henselization of the local ring $\mathbb C[x,y]_{(x,y)}$ . Let $\hat R$ be the $\mathfrak m$-adic completion of $R$. Then there is a natural map $R \to \hat R$ which ...
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### Local rings in the etale topology are henselizations?

I am reading Milne's lecture notes on etale cohomology and am now in the chapter on local rings in the etale topology. Let $X$ be some algebraic variety over an algebraically closed field $k$. Milne ...
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### Prove that for local rings $R,S$, $M_m(R)\cong M_n(S)\implies R\cong S$ and $m=n$

Prove that for local rings (not necessarily commutative, and local means the nonunits form a two-sided ideal) $R,S$, $M_m(R)\cong M_n(S)\implies R\cong S$ and $m=n$. My first attempt was to consider ...
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### Dimension of a certain finitely generated quotient module over a local ring.

I've been stuck on the following question from dimension theory in commutative algebra. Let $(A,m)$ be a local ring and $M$ a finitely generated $A$-module. Given $x_1,...,x_r \in \mathfrak{m}$, ...