# Tagged Questions

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### An explict description of the integral closure of $A=k[x,y]/\langle x^3-y^2\rangle$. [duplicate]

Let $k=\mathbb C$ and $A=k[x,y]/\langle x^3-y^2\rangle$. Denote by $X$ and $Y$ the cosets of $x$ and $y$ in $A$. Question: How do we see that the integral closure $A'$ of $A$ is $k[Y/X]$? Since ...
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### Lie rings: reference request

Dear friends: I am looking for a modern reference for Lie rings (In particular, I would like to have nice references for the structure of Lie ideals), let it be lecture notes or a book, in the sense ...
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### The family of schemes $\operatorname{Spec} A[x]/(x^n)$

Consider the family $S_n:=\operatorname{Spec} A[x]/(x^n)$ of schemes, $A$ denoting any ring (which in our subject always means commutative and with identity). Is there some intuitive picture for ...
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### roots of cubic polynomial

On page 26 of Milne's Elliptic Curves (http://www.jmilne.org/math/Books/ectext5.pdf), he states the following: "... a cubic polynomial $h(x) \in k[x]$ with two roots in $k$ has all of its roots in ...
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### Generalization of Chinese Remainder Theorem to infinite ideals

I'm looking for any (obviously weaker) generalization of this famous theorem in the special case that the family of ideals is not finite.
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### Does This Ring have a Name?

Let $M_1=\{0,1,2,4,5,8,9,10,\cdots\}$ be the set of nonnegative integers that can be written as a sum of two perfect squares. Let $M_2=\{\sqrt{m}: m\in M_1\}=\{0,1,\sqrt{2},2,\sqrt{5},\cdots\}$. Let ...
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### Quasi-hereditary algebras

Can anyone please recommend a reference (I prefer a book chapter) on Quasi-hereditary algebras? and if it is possible tell me how to prove that Schur algebras are Quasi-hereditary (or any other ...
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### Correspondence theorem for rings.

Could someone provide a reference that includes a full and honest proof of the Correspondence Theorem for rings? Let $A$ be a multiplicative ring with identity and $I$ an ideal of $A$. There is a ...
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I'm studying the theory of Noether but I have only 4 pages of lecture notes with no details or examples. Are there any good lecture notes or chapters you know about? In my lectures the basics of ...
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### A criterion for finite generation of subalgebras of a polynomial ring

In her 1926 paper on invariant theory, Emmy Noether uses a certain "finiteness criterion" which I wish to translate and maybe find a more modern reference to. The original wording is: Ein ...
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### How 'commutative' can a non-commutative ring be?

Let $R$ be a finite non-commutative ring. Let $P(R)$ be the probability that two elements chosen uniformly at random commute with each other. Consider the value $$S=\sup_RP(R)$$ where the supremum ...
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### Amenable group rings embeddable in skew fields

I'm looking for a reference of the following fact: given a (countable?) amenable group $G$ and a (skew) field $K$, the following are equivalent: (1) the group ring $K[G]$ is a domain; (2) $K[G]$ is ...
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### Reference request: Morita contexts

During an independent study I've come across Morita contexts, but I'd like to understand them better. A quick Google search doesn't yield much fruit, so I was hoping to find a good reference on the ...
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### An explicit $\Lambda_R^\ell(M)$ when $M$ is not free

Let $\Lambda_R^n(M)$ be the nth exterior power of an $R$-module $M$. Let us assume $M$ is finitely generated. When $M$ if free, say, $M=R^{\oplus d}$, we know \Lambda_R^n(M)\cong ...
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### Subring of $R$ with fraction field $=$ Frac $R$

Let $R$ be an integral domain with fraction field $K$. Of course all the overrings of $R$ share the same fraction field $K$ (by an overring I mean a subring $S\subset K$ containing $R$ as a subring). ...
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### Abstract algebra book recommendations for beginners.

I am taking abstract algebra in this semester. I find that my lecturer didn't provide enough examples for understanding. So I would like to ask what are the recommended books which has lots of solved ...
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### Why is $S/R$ a ring extension?

If $S$ is a ring and $R \subset S$ is a subring it's common to write that $S/R$ is an extension of rings. I frequently find myself writing this and read it quite often in textbooks and lecture notes. ...
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### Epic maps in the category of commutative rings with identity.

Here all rings are assumed to lie in the category $\cal C$ of commutative rings with identity, and ${\cal C} (\ R\ ,\ S\ )$ is the set of all ring homomorphisms $F$ from $R$ to $S$ for which ...
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### The Picard-Brauer short exact sequence

It seems to be a rather well understood fact that, given commutative rings $R,S$, and a homomorphism $R \to S$ there is a short exact sequence \text{Pic}(R) \to \text{Pic}(S) \to F_0 \to ...
### How can I calculate the radical of an ideal in ring ${\Bbb Z}_n$?
I learned the concept radical of an ideal from this wikipedia article. I tried some examples and I found that it's not easy to find $Rad(I)$. (That article gives some examples when $R={\Bbb Z}$.) For ...
### Direct limits and $\rm Hom$
I read that $\lim\limits_{\longleftarrow}\mathrm{Hom}(N_j,M)\cong\mathrm{Hom}(\lim\limits_{\longrightarrow}N_j,M)$. I was wondering if we can write $\lim\limits_{\longrightarrow}\mathrm{Hom}(N_j,M)$ ...