# 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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### Exercise about saturations of a ring

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 ...
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