Tagged Questions

Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, among other topics.

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Small categories

Why $R$-Mod is a small category? There is a way to recognize small categories? For example Grp (i.e. category of all groups) is large because every set can be equiped with a group structure.
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Following groups are not pairwise isomorphic in spite of having the same order? [on hold]

Can someone help me with the following question? "Prove (Z8, +), (D4, ◦) (the group of symmetries of the square) and the quaternion group (Q, ·): Q = {1, −1, i, −i, j, −j, k, −k} are not pairwise ...
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How can I show that $G$ is non abelian of order 20?

Problem says: Let $G=\langle x,y|x^4=y^5=1,x^{-1}yx=y^2\rangle$. Show that $G$ is nonabelian group of order 20. To show it, I tried to turn $x^ny^m$ into $y^kx^l$ for some $k,l$. Since I have ...
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Tensor product of complexes

Let $A$ be a ring and let the modules that are involved be left and right $A$-modules (not necessarily bimodules over $A$). I'll denote as $\mathcal{E}^n_R(M, N)$ the category of $n$-fold extensions ...
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For $H \leq G$, showing that $N_G(H)/C_G(H) \leq \text{Aut}(H)$

This question probably has a very simple answer! I'm trying to understand the proof of the following result from Dummit and Foote, 3ed: Here is the proposition referenced: I don't understand ...
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Is this true: $A \le N_G(B) \not \Rightarrow B \trianglelefteq A$?

Let $G$ be a group, let $A, B$ be subgroups of $G$, and assume $A \le N_G(B)$. My question comes from reading Dummit and Foote, $\S 3.3$: The Isomorphism Theorems. We are proving the Second/Diamond ...
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Finding the maximal ideals of the quotient of a polynomial ring by an ideal

Let the field $k$ be algebraically closed, let $f(X) \in k[X]$ be a separable polynomial of degree at least $2$, let $$B = \frac{k[Y,X]}{(Y^2 - f(X))}$$ and write $y,x$ for the images in $B$ of $Y$ ...
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Importance of universal enveloping algebras and Poincare-Birkhoff-Witt theorem.

Let $L$ denote a Lie algebra and $U(L)$ denote its universal enveloping algebra. I am trying to see why universal enveloping algebra and PBW theorem are important. Precisely: Given any ...
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The square roots of different primes are linearly independent over the field of rationals

I need to find a way of proving that the square roots of a finite set of different primes are linearly independent over the field of rationals. I've tried to solve the problem using ...
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Artinian - Noetherian rings and modules suggest study guide

What text or any document that has gathered this part of Algebra theory. Thanks. Pd: I seek on variety's book of commutative algebra but the subject is partially dealt
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Is a ring R, modulo an ideal I (generated by x), then modulo an ideal J (generated by n) the same as R modulo the ideal generated by (n,x)?

Is the following statement true? $$R/(x,n) = \left[ R/(x) \right] / (n)$$ My thinking behind it was as follows: \begin{array}{ccc} \left[ R/(x) \right] / (n) & = & \{ r+(n) : r \in R/(x) ...
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Onto Group Homomorphism S3 to K4

Problem: Determine if there exists an onto group homomorphism $\alpha: S_3 \rightarrow K_4$ (the Klein Group) Let K be the kernel of alpha. Here is what I have so far: Since any onto homomorphism ...
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a conjugate of a glide reflection by any isometry of the plane is again a glide reflection

Q : Prove that a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, and show that the glide vectors have the same length. I know that: the conjugate ...
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First isomorphism Theorem and Cosets of K

let $\alpha:G \rightarrow G_1$ be a group homomorphism with ker $\alpha$ = K. For $a \in G$ show that Ka = {$g \in G$|$\alpha(g) = \alpha(a)$}. I am studying the first isomorphism chapter of my book ...
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Fundamental theorem of finitely generated abelian groups query

Just had a question about how to apply this theorem. If I am only told that a group is of finite order and is Abelian can we use this theorem? Is there a way to ensure it is finitely generated, or do ...
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$G = \mathbb{Q} / \mathbb{Z}$ surjective map and kernel isomorphism

Let $G = \mathbb{Q} / \mathbb{Z}$, written additively. For all $n > 0$ how come $p_n(x) = nx$ is a surjective homomorphism from $G \rightarrow G$ and how come the kernel of $p_n(x)$ is isomorphic ...
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Can we always find a primitive element that is a square?

Let $L/\mathbb Q$ be a finite field extension. The Primitive Element Theorem says that there is an element $\alpha \in L$ so that $L=\mathbb Q(\alpha)$. Can I always find an element $\beta \in L$ so ...
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Isomorphism of Subgroup of $D_n$

If $k|n$ , $k \ge 2$ I am trying to show that $D_n$ has a subgroup isomorphic to $D_k$. I know that by Lagrange's theorem If a subgroup of order $k$ exists in $D_n$ it will divide $2n$ then $k$ ...
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Quick way to show that inclusion is a local property? [duplicate]

I have encountered a problem which requires me to prove that ideal inclusion is a local property. That is to say, suppose $S,T \subset R$. Show that $S \subset T$ if and only if $SR_P \subset SR_P$ ...
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For $a_{i}:=\sqrt{2+a_{i-1}}$, how to prove that $\mathbb{Q}\subset\mathbb{Q}(a_i)$ is cyclic of degree $2^i$?

Let us define numbers $a_i\in\mathbb{R}_{\geq0}$ by $a_0=0$ and $a_{i+1}=\sqrt{2+a_i}$. How do I prove that $\mathbb{Q}\subset\mathbb{Q}(a_i)$ is cyclic of degree $2^i$? How do I prove that ...
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Prove that a free group of rank $\ge2$ is centerless and torsion-free

This exercise is from Rotman's Introduction to the Theory of Groups. It's just as the title states: prove a free group of rank $\ge2$ is centerless torsion-free. Here, the definition of a free group ...
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Galois theory for beginners, mistake?

I read this short article http://www.math.jhu.edu/~smahanta/Teaching/Spring10/Stillwell.pdf. The goal of this article is to show the unsolvability of certain polynomials of degree $\geq 5$. But at the ...
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If M+N and M$\cap$N are finitely generated modules, so are M and N.

The question asks to prove that if $M+N$ and $M \cap N$ are finitely generated modules, then M and N are also finitely generated. I've tried to use basic definitions, but all failed. I set some ...
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$M + N$ and $M \cap N$ are finitely generated $A$-modules implies $M$ and $N$ finitely generated using exact sequences [duplicate]

Let $0 \to M_1 \to M \to M_2 \to 0$ be an exact sequence of $A$-modules. i) Prove: If $M_1$ and $M_2$ are finitely generated, then $M$ is too. ii) Let $M$ and $N$ be sub-modules of an $A$-module ...
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Show that $|\Bbb Z_n^*|$ is even if $n\ge 3$.

I am trying to show that $|\Bbb Z_n^*|$ (i.e the elements of $\Bbb Z_n$ relatively prime to n forming a group under multiplication) is even if $n \ge 3$ using a corollary in my textbook to Lagrange's ...
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Submodules finitely generated if sum and intersection are finitely generated? [duplicate]

Let $M,N$ be submodules of an R-module $L$, where $R$ is a commutative ring with unity. I would like to prove that if $M+N$ and $M \cap N$ are finitely generated so are $M$ and $N$. Trying to combine ...
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Fibers of extension of scalars to algebraic closure of an affine variety is of dimension 0.

Let $X$ be an affine variety over $k$ and $\overline{k}$ be the algebraic closure of $k$. Then the projection morphism $X_\overline{k} = X \times _k \overline{k} \to X$ has dimension 0 fibers. ...