# Questions tagged [localization]

For questions regarding the process, consequences, and stability of localizing algebraic structures such as rings, categories, and modules. Not for use with local topological spaces.

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### Geometric meaning of localization with an example

My task is to understand the following example: Let $R=\mathbb{R}[x,y]/(xy).$ Let $p=(x-1)$ be the ideal of $R$. Show that $(x-1)$ is a maximal ideal of $R$ and deduce that the localization at this ...
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### Geometric intuition about and examples of locally cyclic modules

A collection $S_1,\dots,S_n$ of monoids (which we always assume contain $1$) in a commutative ring $A$ are said to be comaximal when any $\langle s_1,\dots,s_n\rangle=1$ for any choice of $s_i\in S_i$ ...
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### Question about the restriction $\tau_n>0$ in the definition of local martingales or localization of processes

I have a question while reading the proof of Lemma 3 from the following link: https://almostsuremath.com/2009/12/24/local-martingales/ The Lemma states that a nonnegative local submartingale is a ...
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Consider the $\mathbb{Z}$-module $\mathbb{Z}[1/6]$ and the module $(2020)\subset \mathbb{Z}$. In an algebra exercise from last year, it was asked how many elements the quotient $\mathbb{Z}[1/6]/(2020)\... 1 vote 1 answer 59 views ### Two affine varieties which are isomorphic, but their projectivisations are not I am learning Algebraic Geometry and came across the following question: Show$V(y-x^3) \cong V(y-x^2)$as affine varieties in$\mathbb{A}^2$. Prove that their projectivisations in$\mathbb{P}^2$are ... 3 votes 1 answer 58 views ### Let$M$be an$R$-module. If$I \subseteq \mathfrak{m}$implies$M_\mathfrak{m} = 0$for all max ideals$\mathfrak{m} \subseteq R$, show$IM = M$Suppose we have a ring$R$and an$R$-module$M$. Suppose we have an ideal$I\subseteq R$such that for all maximal ideals$\mathfrak{m} \supseteq I$we have$M_\mathfrak{m} = 0$. I need to show that ... 1 vote 1 answer 74 views ### Very basic question about localizations Let$A$be a ring,$S$a multiplicatively closed subset. Is it true that$\frac a1\in S^{-1}A$is a non zero-divisor if and only if$a\in A$is? I would say yes: for, if$\frac a1$is a zero-divisor ... 0 votes 2 answers 92 views ### Question about universal property of the localization I have this doubt, that arise quite often when I work with localizations. Let$S\subset A$be a multiplicatively closed set of a ring, and let$i:A\to S^{-1}A$be the canonical homomorphism. Suppose ... 1 vote 1 answer 50 views ### localization in the D + M construction Let$T$be an integral domain, and let$K$be a subfield of$T$and$M$a maximal ideal of$T$such that$T = K + M$. Let$D \subseteq K$be an integral domain, and set$R = D + M$. My question is: if ... -1 votes 1 answer 54 views ###$R_P\cong \mathbb{R}[x]_{(x-1)}$Let$R=\mathbb{R}[x,y]/(xy)$and$P\subset R$be the ideal generated by$x-1$. Show that $$R_P\cong \mathbb{R}[x]_{(x-1)},$$ where$R_P$is the localization of$R$at$R\setminus P$(since$R/P$is a ... 2 votes 0 answers 39 views ### Another proof that minimal primes are made of zero-divisors Let$A$be a ring, let$D$be the set of the zero-divisors and$S:=A-D$. Show that any minimal prime ideal of$A$is contained in$D$. I know that this question has already been asked, but (in the one ... 1 vote 1 answer 27 views ### Characterization of maximal multiplicatively closed subsets Let$A$be a non-zero ring, and let$\Sigma $be the set of the multiplicatively closed subsets properly contained in$A$. Show that$\Sigma$admits a maximal element respect to inclusion. Then prove ... 0 votes 0 answers 14 views ### Isomorphism between localizations (2) [duplicate] Let$f:A\to B$be a ring homomorphism, and let$S\subset A$be a multiplicatively closed set. Denote with$T$the image of$S$in$B$. Show that$S^{-1}B\cong T^{-1}B$as$S^{-1}A$-modules. First I ... 0 votes 0 answers 44 views ### Isomorphism between localizations Let$S,T$be two multiplicatively closed subsets of a ring$A$. Let$U$be the image of$T$in$S^{-1}A$. Show that$(TS)^{-1}A\cong U^{-1}(S^{-1}A)$.$S^{-1}A\to (TS)^{-1}A:\frac as\mapsto \frac as$... 1 vote 0 answers 68 views ### Example of a certain ring I have to find a ring$A$that is not an integral domain, but such that$A_p$is an integral domain for every prime ideal$p\subset A$. Set$A:=\mathbb Z/6\mathbb Z$. Its only (prime) ideals are$p:=3\...
Let $A$ be a ring, $S$ any multiplicatively closed subset. For an ideal $I$, let $S(I)$ denote the contraction of $S^{-1}I$ in $A$. If $I$ has a primary decomposition, show that the ideals of the form ...