# Questions tagged [abstract-algebra]

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### $\operatorname{Hom}_k(R,k)$ is injective indecomposable $R$-module for $R$ local $k$-algebra of finite $k$-dimension

I am working on Exercise 3.1.22. from Bruns and Herzog's Cohen-Macaulay rings. The exercise in question is the following (the references I will use are of course from within the book): It appears ...
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### A subgroup with finitely many conjugates which has finite index in its normal closure is finite

I was wondering if the following statement is true: let $G$ be a group and $H$ a subgroup of $G$ such that $H$ is almost normal in $G$ (i.e. $H$ has finitely many conjugates in $G$ or equivalently the ...
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### Does the monoid of non-zero representations with the tensor product admit unique factorization?

Let $(M, \cdot, 1)$ be a monoid. We will now define the notion of unique factorization monoid. A non-invertible element in $M$ is called irreducible if it cannot be written as the product of two other ...
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### Is the direct product of finitely-generated groups cancellative? [duplicate]

The direct product is cancellative for finite groups, so I wanted to know if this result holds for finitely-generated groups as well. The proof linked clearly doesn't apply there, but I have been ...
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### Clarification on Field Homomorphisms

Let $F, k$ be ordered fields. It is clear that a homomorphism $\phi : F \to k$ satisfies the properties of a ring homomorphisms, that is, preserving operations and multiplicative identity. But is it ...
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### How to determine $\Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4,\Gamma_5$?

I started learning about Dirichlet Characters. Here is what I learned so far: Definition: Let $m \in \mathbb{N}$. We call a function $\chi:\mathbb{Z} \rightarrow \mathbb{C}$ a Dirichlet Character mod ...
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### Direct sum of free abelian group and quotient of abelian group by subgroup

I'm currently studying abelian groups in Kurosh's The Theory of Groups. I'm trying to understand the proof of the theorem: Let $B \leqq A$ be abelian groups. If $A/B \cong C$ and $C$ is a free group, ...
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### Is the localization of a zero-dimensional ring a quotient?

If $R$ is any commutative zero-dimensional ring and $m$ is a maximal ideal, then is $R_m$ always naturally a quotient of $R$? In other words, is the natural map $R\to R_m$ always surjective? I was ...
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### Profinite completion of profinite groups

I was trying to prove that $\mathbb{R}/\mathbb{Z}$ cannot be a galoisgroup for any extension. My plan for this was to show that $\mathbb{R}$ is a divisible group and that every quotient of a divisible ...
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### What do the words "descends" and "Induced" mean in the following quoted passage?

The following is taken from pg 4 section 6.1 of the following notes Background $\quad$ We start by observing that a ring homomorphism descends to quotient rings whenever the image of an ideal on the ...
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### Comparing mathematical objects by the "rigidity" of their definitions

A loose interest of mine recently has been ordering mathematical objects by how "combinatorial" their study is, in broad terms. I consider the study of a mathematical object more “...
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### When a quotient of $\mathbb Z_p[[x,y]]$ is a regular local ring

Consider the power series in two variables with p-adic coefficients. Namely consider $\mathbb Z_p[[x,y]]$. Moreover let $c\in\mathbb Z_p$ and construct the quotient: $$W=\mathbb Z_p[[x,y]]/(xy-c)$$ ...
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### Projective modules over group rings and trace

Consider a finite extension of complete valued fields $L/K$ with corresponding extension of rings $B/A$. Suppose $L/K$ is Galois with group $G$. I want to show that $B$ is a projective $A[G]$ module ...
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### Confusion about rings and ideals and how ideals are always subrings.

I've been studying rings and ideals recently and was told that all ideals are subrings. However, I seem to have been able to do the following to show that leads to an ideal being the ring itself. That ...
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### Group homomorphism between multiplicative groups of fields

Let $\mathbb{K}$ and $\mathbb{F}$ be two (algebraically closed) fields. I don't know what a group homomorphism $\mathbb{K}^*\to \mathbb{F}^*$ would look like. It is easy to see that $x\mapsto x^n$ ...
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### Method to find algebraic elements

Which of following is true for x$\epsilon$ R $x^3$ is algebraic over Q implies x is algebraic over Q There is an x such that x is algebraic over Q($\sqrt 2$) but need not be algebraic over Q There ...
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