# Tagged Questions

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### Even XOR Odd Infinities?

Modular Arithmetic (MA) has the same axioms as first order Peano Arithmetic (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$. ...
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### Flatness and local freeness

The following statement is well-known: $A$ a commutative Noetherian ring, $M$ a finitely generated $A$-module. Than $M$ is flat if and only if $M_{\mathfrak{p}}$ is free for all $\mathfrak{p}$. My ...
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### Invariance of $Z[x]$ under a self-equivalence of the category of commutative rings with 1.

Let $\mbox{Rings}$ be the category of commutative rings with $1$. Is there an equivalence of categories $F: \mbox{Rings} \to \mbox{Rings}$ s.t. $$F(\mathbb{Z}[x])\not\cong \mathbb{Z}[x]?$$
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### Axiomatic definition of integers

The real numbers can be axiomatically defined (up to isomorphism) as a Dedekind-complete ordered field. What is a similar standard axiomatic definition of the integer numbers? A commutative ordered ...
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### Is the sheaf of smooth functions flat?

Let $X$ be a smooth algebraic variety over $\mathbb{C}$. Is the sheaf of smooth functions on $X$ flat as an $\mathcal{O}_X$ module?
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### Are these rings of functions isomorphic?

Let $R$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are continuous outside $(-1,1)$ and let $S$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are ...
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### The number of ideals in a ring

Here is a question that I first asked in math.stackexchange, but I think the question must be proposed here. Let $R$ be a finite commutative ring with identity. Under what conditions the number ...
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### Surjectivity of the natural map of injective module to its localization

Hello all, The lemma 3.3 page 214 in Hartshorne Algebraic Geometry book states:" If $I$ is an injective module over a Noetherian ring $A$. Then for any $f\in A$, the natural map of $I$ to its ...
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### When does End(M) consist entirely of zero, zero divisors, and units?

Let $R$ be a commutative ring (with $1$) such that every non-zero divisor in $R$ is a unit (see Rings in which every non-unit is a zero divisor for various stabs at what these are called). Let $M$ be ...
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### finiteness dimension

$R$ is a local Noetherian ring. $f_I(M)$, the finiteness dimension of a module $M$ relative to $I$, is defined in ...
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### relation between Ass Ext(M,N) and Ass M ,Ass N

Let $R$ be a noetherian ring and $M$ , $N$ be finitely generated $R$-modules. Then what is the relation between $Ass\ Ext^i_R(M,N)$ and $Ass\ M, Ass\ N$? $Ass$ means set of associated prime ideals. ...
### chain of prime ideals in polynomial ring $S=\Bbb{R}[x_1,x_2,…,x_n,…]$
Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. Is the following question true? Question: Is there any maximal ...