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

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Writing $I= (xz-y^2, yt- z^2)$ as an intersection of prime ideals

I need to write the ideal $I= (xz-y^2, yt- z^2) \subset R = \mathbb{K}[x,y,z,t]$ as intersection of prime ideals. Any idea? For the moment, I've noticed that $I$ is radical, then it suffices to ...
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isomorphism classes of representations of a quiver

Classify all isomorphism classes of representations of dimension vector 1 and 2 of the following quiver The professor briefly did the solution, but I could not understand what was going on. What he ...
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defining the group law on elliptic curves in general

Let $k$ be an arbitrary field and $C \subset \mathbb{P}^2(k)$ an elliptic curve. In order to define the group law on $C$ we need to establish some geometric facts first, e.g. Any line intersects $C$ ...
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About Rees homomorphism

I am came across the notion Ress Congruence for semigroups. They define it as $$\rho_I=(I\times I)\cup {1_S}$$ wherein $I$ is an ideal of semigroup $S$ satisfying ...
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The alternating group is generated by three-cycles

Prove that, for $n \geq 3$, the three-cycles generate the alternation group $A_n$ Proof: We multiply on the left by 3-cycles to "reduce" an even permutation $p$ to the identity, using induction ...
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A Book for abstract Algebra

I am self learning abstract algebra. I am using the book Algebra by Serge Lang. The book has different definitions for some algebraic structures. (For example, according to that book rings are defined ...
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What do group automorphisms fix?

I have often found it useful to sit and contemplate what kinds of elements, subsets, or structures do the automorphisms of an object fix or permute. Sometimes the observations do not have immediate ...
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Is an embedding of any group into itself always an automorphism?

I came across a question in chapter-8 The power of homomorphism (Visual group theory Book) which says that: Is an embedding of any group into itself always an automorphism? (Hint is that It is true ...
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homomorphism $\phi$ : $\mathbb Z$ $\rightarrow$ $\mathbb Z$ using $\phi(n)$=$2n$

A question from Visual group theory says : consider the homomorphism $\phi$ : $\mathbb Z$ $\rightarrow$ $\mathbb Z$ by $\phi(n)$=$2n$,where the group operation on $\mathbb Z$ is '+'. Would $\phi$ be ...
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34 views

Every subgroup of a cyclic group is characteristic (using lattice theory).

I want to show for $n\in\Bbb N$, which is not square-free, that every subgroup of $Z_n=\langle x\;|\;x^n\rangle$ is characteristic. But I want to show it in a convoluted way. Every automorphism ...
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38 views

Direct Sum on Homology

I have a big problem and i don't know how to solve it i have no idea So, let $i_2: X_2\rightarrow X$ an inclusion and $j_1: X\rightarrow (X,X_1)$ we have that $i_{2_*}: H_k(X_2)\rightarrow H_k(X)$ is ...
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Proof of the theorem : Any $R$-module $M$ is in $R$-injective module

Note that this an exercise in Floote and Dummit's $algebra$ book. Problem : Let $1\in R$. Any $R$-module $M$ is in $R$-injective module Definition : $R$-module $Q$ is $injective$ if $0\rightarrow L ...
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44 views

My first proof related to subspaces (vector spaces). Please comment.

What do you think about my first proof which deals with subspaces? Theorem An intersection of subspaces is a subspace. Preliminaries Corresponding to the notation in Wikipedia, symbols for vectors ...
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411 views

Difference between “space” and “algebraic structure”

What is the difference between a "space" and an "algebraic structure"? For example, metric spaces and vector spaces are both spaces and algebraic structures. Is a group a space? Is a manifold a space ...
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In $\mathbb{Z}/(n)$, does $(a) = (b)$ imply that $a$ and $b$ are associates?

[Update: Based on the hints provided by @zcn and @whacka, I believe I have found a solution. See my answer below.] Below, $R$ is a commutative ring with $1$. In John J. Watkins' Topics in ...
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Subtlety in Correspondence Theorem for Rings

I have something of a subtle question about the correspondence theorem for rings. The theorem is typically stated like this: Let $A$ be a ring, and $I$ an ideal of $A$. There is a $1-1$, ...
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Reputation of mathematical journals [on hold]

For a young mathematician what is most convenient for his reputation publishing in "Expositiones Mathematicae" or in "American Mathematical Monthly"?
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285 views

How many elements are there in the group of invertible $2\times 2$ matrices over the field of seven elements?

How many elements are there in the group of invertible $2\times 2$ matrices over the field of seven elements? Sorry I have no idea so nothing to say? Any clue!
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An element of $SL(2,\mathbb{R})$

Find the relationship between an elliptic element of $SL(2,\mathbb{R})$ and rotation.. An element $A$ of $SL(2,\mathbb{R})$ is called an elliptic element if $|\text{tr}(A)|<2$ As ...
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Cardinality of $m\Bbb Z_n = \{\overline {ma} : a \in \Bbb Z_n\}$

Let $m,n \in \Bbb Z^+$ such that m divides n. I'm trying to find the cardinality of $m\Bbb Z_n = \{\overline {ma} : a \in \Bbb Z_n\}$. So, I think #$(m\Bbb Z_n)= \frac n m = k$. I tried to prove by it ...
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What is the number of subgroups of order $7$?

$G$ be a simple group of order $168$. What is the number of subgroups of order $7$?
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Hilbert series and Cohen-Macaulay ring

Given a series, how can I find a ring which has exactly that Hilbert series? I know only a way, which in particular computes a lexicographic ideal. I need to solve this exercise: Find two rings ...
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19 views

Irreducible Representations and Direct Sums

I am learning about representation theory, and my professor stated the following as a remark: Let $A$ be a $k$-algebra. Every finite dimensional representation of $A$ is a direct sum of ...
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Square-free factorization of polynomials over finite fields

For any $f\in\mathbb{F}_q[X]$, I want to derive an algorithm which computes a factorization $$f=\prod_{i=1}^kf_i^i\tag{1}$$ with square-free polynomials $f_i$. My Ideas: If $f'=0$, we're done ...
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Why $|G|$ even implies $|A(G)|$ also even?

Let $G$ be finite group with even order. Why has the set $A(G)=\{g\in G: g\neq g^{-1}\}$ an even number of elements?
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Prove that $a^n \cdot a^m = a^{n+m}$

Let $a$ be an element of a group $G$. Prove that $a^n \cdot a^m = a^{n+m}$ for any integers $m,n \in \Bbb Z$.
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Dummit and Foote as a First Text in Abstract Algebra

I'm wondering how Dummit and Foote (3rd ed.) would fair as a first text in Abstract Algebra. I've researched this question on this site, and found a few opinions, which conflicted. Some people said ...
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proof that a continuous additive homomorphism $\mathbb{R}^n\to\mathbb{R}^m$ is $\mathbb{R}$-linear

How can we prove that a continuous additive homomorphism $ \Phi \colon \mathbb{R}^{n}\to \mathbb{R}^{m} $ is $\mathbb{R}$-linear. i.e. satisfies $ \Phi (rv)=r \Phi (v)$ for $ r\in \mathbb{R} $ and $ v ...
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49 views

Does there exists a homomorphism for any groups $G$ and $H$

This is a question from Exercise 8.2 of Visual Group Theory which says:determine whether true or false. For any group $H$ and $G$,there is some homomorphism from $H$ to $G$. For any groups $H$ and ...
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A group of order $120$ has a subgroup of index $3$ or $5$ (or both)

What I have tried that number of $2$-sylow subgroup can be $1,3,5$ or $15$.I have solved the problem when the number of $2$-sylow subgroup is $1,3,5$. But I am not able to solve it for $15$. Any help ...
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Isomorphism of $\mathbb{Z}/n \mathbb{Z}$ and $\mathbb{Z}_n$

Let $n\in \mathbb{Z}^+$. How do I prove that $\mathbb{Z}/n\mathbb{Z}$ is isomorphic to $\mathbb{Z_n}$? Is there any good homomorphism $\phi$ I could use that graphs $\mathbb{Z}/n\mathbb{Z}$ to ...
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Examples for infinite Hamiltonian group

During teaching some basic concepts about a Hamiltonian group, I was asked about an infinite sample. According to what D.J. Robinson cited, we have a very good ...
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Examples of reduced associative algebras

An associative $K$-algebra A is called reduced if $A/rad(A)$ has no nilpotent elements. It can be shown that this is equivalent to that $A/rad(A)$ is a isomorphic to a direct sum of division algebras. ...
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smash product of Eilenberg-Maclane spaces

Let $G$ be an abelian group and $K_n=K(G,n)$ be the Eilenberg-Maclane space. How to obtain $K_m\wedge K_n$ is $(m+n-1)$-connected? (Hatcher's book page 404)
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Is $\cos \frac{5\pi}{8}+i\sin\frac{5\pi}{8}$ in $U_8$, the multiplicative group of the $8$th roots of unity in $\mathbb{C}$?

I encountered this problem while reading Fraleigh, A First Course in Abstract Algebra, 4/e. (p.60 #19) Let $U_8$ be the multiplicative group of the $8$th roots of unity in $\mathbb{C}$. The question ...
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About automorphisms of commutative semigroups

Suppose that $M$ is a commutative monoid and that the product $P$ of $M$ and the nonnegative integers $\mathbb{N}$ with addition has no nontrivial automorphisms. The set $S$ of pairs $(m,n)$ in $P$ ...
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Are there counter intuitive interpretations of ZF or ZFC?

Are there interpretations of ZF or ZFC that are non standard in the sense that $\epsilon$ is interpreted in a counter intuitive way that intuitively has nothing to do with "belongs to" or "is part of" ...
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Prove that the symmetric group $S_n$ is not abelian for $n \geq 3$

I am trying to prove this: Let $\sigma$ be a non-identity element of $S_{n}$. If $n \geq 3$ show that $\exists \gamma \in S_{n}$ such that $\sigma\gamma \neq \gamma\sigma$. Hint: Let ...
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When is a vector space (over field $K$) also a ring (with subring $K$)?

(Apologies in advance for the very naive question. I'm just learning about all this. Also, for the sake of expedience, below I use the word "ring" when it would more correct for me to use ...
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A solution of a linear system in some extension field implies a solution in the subfield

Fix a field extension $k\subseteq K$ and consider a linear system $Ax=b$ where $A$ is a matrix (not necessarily square) with coefficients in $k$. I don't understand why if the above linear system ...
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33 views

Left Cosets of Cyclic Subgroup

Question from a GRE Math book that I'm having trouble understanding: Find the number of left cosets of the cyclic subgroup generated by (1, 1) of $$Z_{2} \times Z_{4}$$ where Zn denotes the cyclic ...
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Reference request for properties of harmonic polynomials

I am reading this paper, and on page $4$ it takes a non-degenerate quadratic form $q$ on a finite dimensional complex vector space $V$, defines the laplacian assosciated to $q$, $\Delta$, acting on ...
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Characterizing kernel of ring morphism

Let $K$ be a field and define a ring morphism $\psi: K[x_1,x_2, \dots , x_n, y_1, y_2, \dots , y_n] \rightarrow K(x_1,x_2, \dots , x_n)$ by $\psi(x_i) =x_i$ and $\psi(y_i) =\frac{1}{x_i}$. I think ...
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homotopy groups of wedge sum

Let $X_\alpha$ be connected CW-complexes. Then from Hatcher's book, $$\pi_{n}(\prod_{\alpha} X_{\alpha})=\prod_{\alpha}\pi_{n}(X_{\alpha}).$$ Is it true in general $$\pi_{n}(\bigvee_{\alpha} ...
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Does every free $R$-module have a maximal proper submodule?

Let $R$ be a commutative ring with $1$. We know that every finitely generated $R$-module has a maximal proper submodule. Is it true for any free $R$-module? In particular, can we do the following: ...
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For an associative binary operation with identity, the set of invertible elements forms a group

Let $S$ be a set, and $*$ an associative binary operation on $S$. Suppose there is an element $e\in S$ such that ($1$) $e*x=x$ and $x*e=x$ for all $x\in S$. (a) Prove that there is a unique element ...
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Why should I consider the components $j^2$ and $k^2$ to be $=-1$ in the search for quaternions?

I'm reading a paper about Hamilton's discovery of quaternions and it explains why he failed in his 'theory of triplets' where he tried to make a vector with $3$ dimensions, as an analogy to the ...
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Show that there is no surjective ring homomorphism from $\mathbb Z_2[x]$ to $\mathbb Z_2 \times \mathbb Z_2\times \mathbb Z_2$

I saw this question as a bonus from a past exam, and here's my solution for verification. I argued like so. I said suppose there is such a surjective homomorphism $f$, then $f(0)=(0,0,0)$, $f(1)= ...
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Commutative semiring such that every element except zero does not have additive inverse and each element can be uniquely sum-decomposed

Is there a unique factorization commutative semiring such that every element except zero does not have additive inverse in the semiring and each element can be decomposed into unique finite sum ...
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Is there a theory of “rings” with partially defined multiplication?

Consider the abelian group $R [[\mathbb{Z}^d]]$ of all formal Laurent series over a commutative ring $R$ (a typical element has the form $\sum_{v \in \mathbb{Z}^d} \lambda_v \cdot X_1^{v_1} \dots ...