# Tag Info

Accepted

### Can an intersection of ideals in a Noetherian ring be replaced by a countable intersection?

Yes it's true. First, in a (commutative) noetherian ring $A$, every chain of ideals is well-ordered by reverse inclusion. The supremum of ordinal types of such chains is denoted $o(A)$. A ...
• 60.2k
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### Inverse of the Structure Theorem for Finitely Generated Modules over PID

There seems to be quite some literature about rings with this property, sometimes under the name "FGC domains". From Googling, not personal knowledge: In Theorem 14 of Kaplansky, Irving, Modules ...
• 33.9k
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### A question on a Macaulay2 computation

Commutative algebra is NOT the same as algebraic geometry, especially projective algebraic geometry. The variety in $\mathbb{P}^9$ defined by $I$ and the variety in $\mathbb{P}^9$ defined by $I_0$ are ...
• 2,868
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### Reduced ring with all non-prime ideals finitely generated

Question: Let $R$ be a reduced ring with all non-prime ideals finitely generated. Then is $R$ Noetherian? The answer is Yes. To lessen my typing, let me use the abbreviation NFG to mean not-finitely-...
• 12.1k
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### Is the ideal product presheaf a sheaf? Do we have any reasons to believe it will be / it won't?

It need not be a sheaf. As an example, consider a space $X$ which is a disjoint union of open subspaces $X_n$, and pick $\mathcal O_X,\mathcal I,\mathcal J$ with the property that some element $c_n$ ...
• 27.4k
Accepted

### $I,J$ are $p$-primary ideals, but $I+J$ is not

Let $R$ be the commutative ring $k[x,y,z]$. Let $I$ be the ideal generated by the regular sequence $(x^2,y)$. Let $J$ be the ideal generated by the regular sequence $(x^2,y-xz)$. Then both $R/I$ ...
Accepted

### Distribution relation in the Euler system of Heegner points

What I don't understand is why the exactly the same terms should appear in both sums. The Galois action on CM points is described in adelic terms via the fundamental theorem of complex multiplication ...
• 10.3k
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### Is every 2-sided ideal in a C*-algebra hereditary?

No. Take $A = C[0,1]$ and let $I$ be the (unclosed) ideal generated by the function $f(t) = t$. This ideal is self-adjoint, but it does not contain the function $g(t) = t\sin^2(\frac{1}{t})$, so it is ...
• 42.1k

### Maximal subideal of an ideal

Claim: Let $T$ be an ideal in a commutative ring. If $T$ contains a unique maximal proper subideal, then $T$ is principal. Proof: Indeed take any element outside the unique maximal proper subideal. ...
• 42.5k

### Do relatively prime polynomials $f$ and $g$ in $k[x,y]$ generate an ideal of finite codimension?

A standard undergraduate maths approach is via resultants. I am not going to survey resultants here (but see below), I'll just say that an immediate consequence of 1) is that there exists a nonzero ...
• 13.7k

### Is the ideal product presheaf a sheaf? Do we have any reasons to believe it will be / it won't?

So here is a counterexample which is qcqs: Take $X$ the affine line with double origin $a_1$ and $a_2$, then take $I_1$ and $I_2$ the ideal of functions vanishing each at one of the origins ...
• 296
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• 4,460
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### On the set of non-zero elements in an integral domain whose generating principal ideal is of a special kind

The answer is yes if $R$ is any atomic domain, e.g., $R$ is a Noetherian domain. Claim 1. Let $R$ be any integral domain. The set $S = S_R$ is saturated in the sense that if $ab \in S_R$ for ...
• 7,463
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### Ideals of Banach algebras

Just to answer the easy part of the question quickly 1) [0,1] 2) the closed unit disc 3) SO(3)
• 25.5k

### A property of minimal prime ideals in commutative reduced ring

This is an interesting question. Quentel's Example provides an example. Let $R$ be a reduced ring, $Min(R)$ its space of minimal prime ideals, and $q(R)$ its classical quotient ring. Theorem The ...
• 279
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### Ideal generated by two univariate, coprime, integer polynomials

For question 2, compute a Gröbner basis over $\mathbb{Z}$ for the ideal generated by $f(x)$ and $g(x)$, which gives the required generator. You can do this easily in SageMathCloud (available free to ...
• 2,738
Accepted

### Ideals invariant under ring automorphisms

There are tons of examples. Put $u = x^2+xy+y^2$ and $v = xy(x+y)$. Then $u$ and $v$ are $SL_2(\mathbb{F}_2)$ invariant. If $f(s,t)$ and $g(s,t)$ are homogenous polynomials with respect to the grading ...
• 151k
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### Is the Scott topology generated by the ideals as the closed sets?

The answer is no. Consider poset consisting of infinitely many incomparable elements $a_1,a_2,\dots$ and a single element $b$ larger than them all. Then $A=\{a_1,a_2,\dots\}$ is closed in the Scott ...
• 27.4k

### Proof in Schertz's Complex Multiplication

Presumably, the author aimed only at showing that the order $\mathfrak{O}_t$ is a one-dimensional Noetherian domain, i.e., it is a Noetherian domain and every of its non-zero prime ideals is maximal. ...
• 7,463
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### Proof in Schertz's Complex Multiplication

The statement that seems odd to you is indeed false, what (I believe) he wants to say is that $\mathfrak{D}_t$ is a Noetherian ring in which every prime ideal $\mathfrak{p}\neq 0$ is maximal. I ...
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### For an integral domain $R$ when does $a^2 \equiv b^2 \bmod 4R$ imply $a \equiv b \bmod 2R$?

Denote $a-b=x$, then $a^2-b^2=x(x+2b)=4z$ for $z\in R$. Assuming that $R$ is integrally closed, we see that $(x/2)^2+b(x/2)-z=0$, so $x/2$ is an algebraic integer, thus $x/2\in R$.
• 103k
Examples include all non-unital algebraically simple $C^\ast$-algebras. By [Blackadar, Bruce E.; Cuntz, Joachim The structure of stable algebraically simple C∗-algebras. Amer. J. Math. 104 (1982), no. ...