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.

learn more… | top users | synonyms (1)

9
votes
1answer
51 views

$M_1 \to M_2 \to M_3 \to 0$ exact iff $0 \to \text{Hom}_A(M_3, N) \to \text{Hom}_A(M_2, N) \to \text{Hom}_A(M_1, N)$ is exact.

Let $M_1 \to M_2 \to M_3 \to 0$ be a sequence of homomorphisms of $A$-modules. Is this sequence exact if and only if the induced sequence of abelian groups$$0 \to \text{Hom}_A(M_3, N) \to ...
2
votes
1answer
29 views

real valued functions with composition

If $G$ is the set of all functions $f : \mathbb{R} \to \mathbb{R}$ such that $f(x) \ge 0$ for all $x \in \mathbb{R}$ with $f ∗ g = f \circ g$ (here $\circ$ denotes the operation of composition), for ...
2
votes
2answers
41 views

Proof that $n\Bbb Z \leq \Bbb Z$ and are the only subgroups of $\Bbb Z$

My challenge is Prove that if $n = 0,1,2,\ldots$ and $n\Bbb Z = \lbrace nk: k \in \Bbb Z \rbrace$, show that $n\Bbb Z$ is a subgroup of $\Bbb Z$ and are the only subgroups. I handled the first ...
7
votes
2answers
92 views

For any rng $R$, can we attach a unity?

Let $R$ be an rng. (There may be no unity) Then, does there always exist a ring(with unity) $A$ such that $R$ is a subrng of $A$?
0
votes
1answer
33 views

I have to show center of $M_n(H)$ is $\mathbb{R}I$ [on hold]

Let $H$ be a real quaternion ring. I have to show that the center of $M_n(H)$ is $\mathbb{R}I$. Can anyone help?
1
vote
2answers
14 views

Proof of the right and left cancellation laws for Groups

I was asked to proof the right and left cancellation laws for groups, i.e. If $a,b,c \in G$ where $G$ is a group, show that $ba = ca \implies b=c $ and $ab = ac \implies b = c$ For the first ...
-1
votes
1answer
17 views

Concavity and quasiconcavity… [on hold]

How do you explain the difference between concavity and quasi concavity? or convexity and quasi convexity?
3
votes
2answers
39 views

Fiber product of groups

Let $f: G \longrightarrow K$, $g: H \longrightarrow K$ be group homomorphisms. For the fiber product $G \times_K H$ to exist, is it necessary to require that $f$ and $g$ be surjective? I can't see ...
1
vote
1answer
13 views

Possible angles between roots in a root system

Given a Root System $\Phi$ let $\alpha,\beta \in \Phi$ with $\alpha \neq \pm \beta$ and $||\beta||\geq ||\alpha||$. Let $\theta$ be the angle between $\alpha$ and $\beta$. Since $<\alpha,\beta> ...
-1
votes
2answers
35 views

Is the subgroup of a non-abelian group is non-abelian?

Is the following statement always true Subgroup of a non-abelian group is non-abelian
2
votes
0answers
38 views

Endomorphism commutes with its adjugate

Let $R$ be a commutative ring, $M$ a free $R$-module of rank $n$ and $f \in \rm{End}(M)$. The adjugate $f^\sharp$ of $f$ is defined by the equalities $$ f^\sharp(x) \wedge y = x \wedge ...
3
votes
2answers
52 views

Proving that disjoint unions of presentations are coproducts of groups

I'm working through Aluffi's Algrebra: Chapter 0 and I need some assistance with an excercise. Aluffi, Ex. II.8.7 Let $(A|R)$, resp. $(A'|R')$, be a presentation for a group $G$ in Grp, resp. ...
2
votes
1answer
25 views

$\operatorname{Rad}(k)=\operatorname{Rad}(L)$

Given a Lie Algebra $L$ on a field $F$, we define the radical of $L$ $\operatorname{Rad}(L)$ as the largest solvable ideal of $L$. We define the adjoint representation ...
4
votes
1answer
47 views

Classifying groups of order 18

I am trying to classify groups of order 18. So far, I have shown that a group $G$ of order 18 is given by $G\cong C_9 \rtimes_{\varphi} C_2$ or $G\cong (C_3 \times C_3)\rtimes_{\varphi} C_2$. If ...
5
votes
4answers
57 views

Prove that $f=x^6+ax+5$ is reducible over $\mathbb{Z_7},\forall a\in\mathbb{Z_7}$

We have $f=x^6+ax+5\in\mathbb{Z_7}$ and we have to show that it is reducible on $\mathbb{Z_7}$, $\forall a\in\mathbb{Z_7}$. Here are all my steps: For $a=0$ we'll get $f=x^6+5\in\mathbb{Z_7}$. But ...
-1
votes
1answer
42 views

Fraction modulo integer in sage [on hold]

I'm working on a sage script right now, I have some polynomials coefficients that are rational, and I want to apply a congruence on these coefficientss, for example: $p = 1 + (7/2)x$ the function ...
0
votes
0answers
26 views

lattices and torsion free sublatices.

I have the following statement that I cant proof, which according to my book is trivial. Let $N$ be a lattice. Let $N_1 \subset N$ be a sublattice such that $N/N_1$ is torsion free. Then it followes ...
1
vote
1answer
17 views

$UTM_n[D]$ is artinian

Why is the upper triangular matrices over a division ring D is artinian? I tried to find properties of this class of rings. The only thing I found that the jacobson radical of this ring is the ...
1
vote
1answer
53 views

To find the no. of elements of order $7$ in a field of 8 elements

Let $F$ be a field of $8$ elements and $A= \{x\in F \,|\, x^7=1 \text{ and } x^k\neq 1 \text{ for all natural number $k<7$}\}$. Then the number of elements in $A$ is: a) 1 b) 2 c) 3 d) 6 ...
1
vote
1answer
34 views

Free module over a set

I think I understand the definition of a free $R$-module over a set $X$: it is given by the set of all maps from $X$ to $R$ which vanish at all but finitely many points of $X$. The module operations ...
4
votes
2answers
71 views

Matrix ring $M_2(\mathbb{C})$, $\mathbb{C}^2$ with $M_2(\mathbb{C})$-module structure.

Let $R$ be the matrix ring $M_2(\mathbb{C})$. Let $M = \mathbb{C}^2$ with its natural $R$-module structure (just given by the usual action of $2 \times 2$ matrices on $2$-dimensional vectors). My ...
3
votes
1answer
29 views

The coproduct of a family of objects of a Preorder (seen as a category)

If the coproduct of a family of objects of a Poset (seen as a category) is the least upper bound, who is the coproduct of a family of objects of a Preorder (seen as a category)? My intuition ...
2
votes
1answer
36 views

Perfect pairing induces isomorphism of tensor products

Let $M, N$ be $R$-modules and $(\cdot, \cdot): M \times N \to R$ be a perfect pairing. Wikipedia sais that this means that the map $\varphi: M \to \text{Hom}_R(N, R), m \mapsto (n \mapsto (m, n))$ is ...
2
votes
2answers
25 views

Cyclic projective module

Let $R$ be an integral domain. If $M$ is a cyclic $R$-module which is also projective, then must there necessarily be an isomorphism of $R$-modules $M \cong R$?
1
vote
1answer
45 views

Relation of $\operatorname{Ext}$ and projective dimension

I have some problem to understand the proof of proposition 8.38, page 473 from An Introduction to Homological Algebra by Rotman. Proposition: Let $x\in Z(R)$ be an element which is not a zero-divisor ...
5
votes
4answers
171 views

Showing that $7+\sqrt[3]{2}$ is an algebraic number

How do I go about showing that $7+\sqrt[3]{2}$ is an algebraic number? I need to show that it is the root of an integer valued formal polynomial? How do I solve these problems in general? I haven't a ...
0
votes
1answer
23 views

Prove that if $A$ is $R$-projective and $C$ is $S$-injective then $\operatorname{Hom}_R(A,C)$ is $S$-injective

In the situation $(_RA,_RC_S)$, prove that if $A$ is $R$-projective and $C$ is $S$-injective then $\operatorname{Hom}_R(A,C)$ is $S$-injective. I appreciate your help.
0
votes
0answers
26 views

Is $\oplus_{i \in I} M_i$ necessarily projective?

If $\{M_i\}$ is a collection of projective modules over a ring $R$, then must the direct sum $\oplus_{i \in I} M_i$ necessarily be projective? I understand that each direct sum of two modules is ...
2
votes
1answer
25 views

If $A \times B$ is the direct product of two rings, must $A$ necessarily be a projective $A \times B$-module?

If $A \times B$ is the direct product of two rings, must $A$ necessarily be a projective $A \times B$-module? I understand I am supposed to think of $A$ as an $A \times B$-module by identifying ...
2
votes
2answers
49 views

Proof that $a\mid x, b\mid x, \gcd(a,b)=1 \implies (ab)\mid x$

I need to prove that: $$a\mid x, b\mid x, \gcd(a,b)=1 \implies (ab)\mid x$$ What I thought was: $$a\mid x \implies x = aq_1\\b\mid x\implies x = bq_1$$ Also, since $\gcd(a,b) = 1$, we have that ...
0
votes
2answers
32 views

Show that $\mathbb{F}_q$ is a Splitting Field for Polynomial $x^q-x$

I have been given a homework problem which asks me to assume that $\mathbb{F}_q$ is a field of order $q$, and to show that $\mathbb{F}_q$ is a splitting field for the polynomial ...
1
vote
0answers
26 views

Show that $I + J = \{a + b\vert a \in I, b \in J\}$ is an ideal of R.

Let $I, J$ be ideals of a ring $R$. Show that $I + J = \{a + b\vert a \in I, b \in J\}$ is an ideal of R. Because $I,J$ are ideals of $R$, so $I,J$ both have $0$, thus $0+0=0\in I+J$. This shows ...
0
votes
1answer
47 views

What kind of algebraic structure is this?

Suppose that over a set are defined two binary operations - "+" and "*", where the first is associative and commutative, and the following law holds: $(x + y) * z = x + (y * z)$ This law is stronger ...
0
votes
1answer
39 views

Are Zero Degree polynomials Considered monics?

DO zero degree polynomials that is constant polynomials considered monic polynomials? Example F(x)=16 Does it Matter the Field or the Integral region where i take the coeficients from?Sorry if the ...
3
votes
2answers
112 views

Basic question about finite fields and characteristic

I am reading Herstein's "Topics in Algebra" and I've encountered with the following problem: If $D$ is an integral domain and $D$ is of finite characteristic, prove that the characteristic of $D$ is ...
0
votes
2answers
54 views

Squares in $\mathbb Z_p$

Let $p\neq 2$, then I want to understand that every element in $\mathbb Z_p$ (p-adic integers) is a square. For the prove one must see that $2$ is invertible in $\mathbb Z_p$. But $2$ is the element ...
0
votes
3answers
106 views

What is the intuition behind the definition of the kernel of a homomorphism

I was starting to study some algebra (groups and homomorphisms in particular) and came across the definition of the kernel (for a group-homomorphism $f:G \rightarrow G'$): $$\ker(f) = \{ x \in G \mid ...
2
votes
1answer
37 views

Why $\zeta^m \alpha \in K[\zeta]$?

In the following lemma from "The Algebraic structures of group rings" : by D.S. Passman, What does $K[G]$ contained isomorphically between $K[\zeta_1, \ldots \zeta_n] $ and $K(\zeta_1, \ldots ...
0
votes
1answer
41 views

Prove that $L=K[x]/\langle m(x)\rangle$ is an algebraic extension of $K$

Let $m(x)$ be an irreducible polynomial of degree $n$. How can I prove that $$L=\frac{K[x]}{\langle m(x)\rangle}$$ is: A vector space of dimension $n$, An algebraic extension of $K$? Edit: I'm ...
4
votes
0answers
36 views

Does every irreducible projective cubic curve have a nonsingular point of inflection?

Does every irreducible projective cubic curve necessarily have a nonsingular point of inflection? I've been trying to construct counterexamples, to no avail, which leads me to believe the question can ...
9
votes
7answers
107 views

Why is $\mathbb{Z}[X]$ not a euclidean domain? What goes wrong with the degree function?

I know that $K[X]$ is a Euclidean domain but $\mathbb{Z}[X]$ is not. I understand this, if I consider the ideal $\langle X,2 \rangle$ which is a principal ideal of $K[X]$ but not of $\mathbb{Z}[X]$. ...
5
votes
3answers
82 views

examples of non-abstract rings?

In group theory, it helped me a lot to use symmetry groups of geometrical objects like a triangle to understand the more abstract concepts. Are there rings with a similar low level of abstractness? I ...
0
votes
1answer
26 views

Polylogarithms and the shuffle algebra

$1)$ Write $\text{Li}_2(1-\frac{1}{x})$ in terms of $\text{Li}_2(x)$ and logarithms by considering its integral representation and suitable changes of variables. Attempt: The di-log is defined as ...
0
votes
0answers
15 views

I.N.Herstein Topics in algebra problem no 2.5.18 [duplicate]

If $H$ is a subgroup of $G$.Let $N= \cap\,\, xHx^{-1} \;\;\;\forall x\in G$. Prove that $N$ is a subgroup such that $aNa^{-1}=N$. I've proved the subgroup part but couldn't show the second part.
2
votes
1answer
41 views

Localising a polynomial ring and non-maximal prime ideal

I'm trying to work out the following past paper question and I've got stuck. $R$ is an integral domain and $S = R[t]$, the polynomial ring in one variable over $R$. We have that $Q$ is a prime ideal ...
0
votes
1answer
35 views

If you have a field isomorphism and the domain is algebraically closed then so is the image?

I know it makes sense because if they are isomorphic they are practically the same thing, but what would a proof look like?
1
vote
4answers
108 views

Show that $P(x,y)=x^6+y^6$ is reducible over $\mathbb{R}$ [on hold]

Show that $P(x,y)=x^6+y^6$ is reducible over $\mathbb{R}$ In general , How do you show that a given polynomial is reducible over some field ?
1
vote
1answer
35 views

The pushout of an epimorphism is an epimorphism

I'm reading the book "Handbook of Categorical Algebra: Volume 1, Basic Category Theory" by Francis Borceux, and in page 52 he states that "..."the pullback of a monomorphism is a monomorphism". ...
1
vote
0answers
17 views

A Lie Algebra $L$ is reductive iff it is completely reducibile as an $\operatorname{ad}_L(L)$-module

Given a Lie Algebra $L$ we say it is reductive if $\operatorname{Rad}L=Z(L)$. How can we prove that $L$ is reductive iff it is an $\operatorname{ad}_L(L)$-module completely reducibile? Suppose $L$ ...
1
vote
2answers
46 views

A basic isomorphism of modules (useful for Corollary 2.7 of Atiyah and MacDonald).

Let $M$ be an $A$-module, $N$ a submodule of $M$, $\mathfrak{a} \subseteq A$ an ideal such that $M = \mathfrak{a}M + N$. Then $\mathfrak{a}(M/N) = (\mathfrak{a}M+N)/N$ I am having troubles in ...