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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exact sequence problem

Let $B_1 \stackrel{f}{\to} B \stackrel{g}{\to} B_2 \to 0$ be an exact sequence. For any module $M$ the sequence $$0 \to \mbox{Hom}(B_2,M) \stackrel{g*}{\to} \mbox{Hom}(B,M) \stackrel{f*}{\to} ...
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Characterize the commutative rings with trivial group of units

This question suggested me the following: Characterize the commutative unitary rings $R$ with trivial group of units, that is, $R^{\times}=\{1\}$. The local case was solved here long time ago ...
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Localization and Direct limit [duplicate]

Let $ A $ be a ring. Let $ I $ be a preordered set, filtering. Let $ \Sigma $ a multiplicative subset of $ A $. Suppose for any given $ i \in I $ a multiplicative subset $ S_i $ of $ A $ contained ...
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Quotient of polynomials, PID but not Euclidean domain?

While trying to look up examples of PIDs that are not Euclidean domains, I found a statement (without reference) on the Euclidean domain page of Wikipedia that $$\mathbb{R}[X,Y]/(X^2+Y^2+1)$$ is ...
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Subgroups of $\mathbf{Z}/20\mathbf{Z}$ using Lagrange's Theorem

According to Lagrange's Theorem, what are the possible sizes of the subgroups of $\mathbf{Z}/20\mathbf{Z}$? I have no idea how to go about answering this. I have a feeling that I should be ...
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Exercise 7.10 Atiyah, $M[x] $ is a noetherian $A[x] $-module [duplicate]

The exercise is: Let $M$ be a noetherian $A$-module. Then $M[x] $ is a noetherian $A[x] $ module. The action of $A[x] $ on $M[x] $ is the obvious one. In a previous exercise it was shown that ...
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Galois closure of $\mathbb{C}(T,\sqrt{T^2+T+1})$ over $\mathbb{C}(T^3)$

I'm trying to solve the following problem. But it's too difficult for me. Let $\mathbb{C}(T)$ be a function field, and put $L:=\mathbb{C}(T,\sqrt{T^2 + T +1})$, $K:=\mathbb{C}(T^3)$. Let $M$ be a ...
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An integral domain with the factorization property and gcd for every two elements is a UFD

Theorem 0.6.1 of Roman's book Field Theory says: Let $R$ be an integral domain for which the factorization property holds (factorization property means that every non zero non unit can be written as ...
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Additive identity in $\mathbb{Z}_{7}$

The problem consists of two parts: 1)Prove that for some integer $m,$ $3^{m} = 1$ in the integers modulo 7, $\mathbb{Z}_{7}$, which i have done. The second part asks to use this fact to deduce that ...
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Does ternary operations have associative property?

Binary Operation is a function. Right? We know that all Binary operations have associative property. They must be either associative or non-associative. The condition is : $$(a*b)*c = a*(b*c)$$ ...
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$R^{(I)} \cong K \oplus H$ where $R^{(I)}$ is free but $K$ is not free

Let $R$ be a commutative ring with unit. Is there an example of a direct sum of $R$-modules $$R^{(I)} \cong K \oplus H$$ where $R^{(I)}$ is free but $K$ is not free ? Clearly $R$ can't be a PID.
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$k[X,Y]/(f)$ not finitely generated as a module (Exercise 4.10 Reid, UCA)

I have been wrestling with this problem for some time and I still can't find $f$. It seems really simple, which annoys me even more. The problem is as follows (Exercise 4.10 Reid, UCA): Suppose ...
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35 views

How many relations exist in the set of A

When A = {1,b,ø}, how many reflexive relations exist on the set? I have said that AxA={(1,1), (b,b), (ø,ø), (1,b), (1,ø), (b,1), (b,ø), (ø,1), (ø,b)} Would I be right in saying that there are only 3 ...
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Regular elements of a module is open and dense

Let $\mathfrak{g}$ be an algebraic Lie algebra and $V$ be a $\mathfrak{g}$-module, then for each $v\in V$, define $\mathfrak{g}^v = \{x\in\mathfrak{g}:xv = 0\}$. Let $V_{reg}$ be the set of all $v$ ...
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Interview preparation for Ph.D admission

I have recently gave a written exam for admission to Ph.D program to an institute in India. I have done that exam well and hoping for an interview call. I would like to know what could be type of ...
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Kahler forms of a smooth affine algebra vanish eventually?

If $k$ is a Noetherian ring, then do the Kahler forms of a smooth affine $k$-algebra of dimension $d$ vanish above $d$? I mean is: $\Omega^{d+1}_{A|k}\cong 0$?
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Classification of separable algebras over a commutative ring

A separable algebra over a field can be classified as a finite product of matrix algebras over division algebras whose centers are finite dimensional separable field extensions of the field. (See ...
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30 views

Nomenclature Question: Algebraic Studies

What are magmas, monoids, groups, rings, modules, fields, algebras, etc. called? I have been calling them "algebraic structures", but I want to know their real/recognized and general name. I would ...
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44 views

Sets as extremely trivial groups

A group is a structure defined upon an underlying set which is endowed with a single binary operator that has some rules attached to it. I was wondering whether one could describe a set itself as ...
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an ideal of matrix ring which is projective

Let $K$ be a field and $$ A=\left\{ \begin{pmatrix} a&b&c\\ d&e&f\\ 0&0&g \end{pmatrix} :a,\dots,g\in K \right\}, $$ then $$ J=\left\{ \begin{pmatrix} 0&0&c\\ ...
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912 views

Is vector space a field? Or more than that?

As you all know, vector space is closed under scalar multiplication, scalar product, vector product and addition. If I take scalar product, vector space is a field, but if i take vector product, ...
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388 views

Subgroups and quotient groups of finite abelian $p$-groups

Motivation The fundamental theorem of finite abelian groups gives us a concise description of the isomorphism types of finite abelian $p$-groups $G$ (in the following, $p$ is a fixed prime). The ...
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A field with characteristic $0$ contains $\mathbb Q$

To prove that a field $F$ with characteristic $0$ contains $\mathbb Q$, the following lemma is used. Lemma: Let $R$ b a ring with unity. If the characteristic of $R$ is $0$, then $R$ contains a ...
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Degree of a single extension of a field. [closed]

Let $K$ be a field. Let $L:=K(b)$ where $b^m\in K$ but $b^i\not\in K$ for all $0<i<m$. Then is $[L:K]=m$?
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Units and Primes in a Ring

Is it true that units in a ring (maybe involves in quaternions) have norm of 1? (norm of 1 does not imply that it is a unit, right?) What about the statement that the number is prime if and only if it ...
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59 views

Solvability of polynomials over fields of characteristic zero

1) Let $K$ be a field, $\operatorname{char}(K)= 0$, and $f ∈ K [x]$ with $\deg(f)\le4$. Then $f$ is solvable by radicals. Proof: $\operatorname{Gal} (F/K) \cong S_4$ then $\operatorname{Gal}(F/K)$ ...
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$\mathbb{Q}[X,Y]/(Y^2-X^3)$ is not a UFD

I'm trying to show that $R = \mathbb{Q}[X,Y]/(Y^2-X^3)$ is not a UFD, but I got stuck. To prove this, I could try to find two "different" factorisations for one element, but I am not familiar with ...
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What are some examples of non-commutative monoids that are both idempotent and self-distributive (on both sides)?

In the presence of the axioms for a commutative monoid, idempotency is equivalent to self-distributivity. Proof. Suppose a commutative monoid is idempotent. Then: $$x(yz) = xxyz = (xy)(xz)$$ On the ...
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Geometric Intuition for Dihedral Group Automorphisms

I noticed the other day that the automorphism group of the dihedral group $D_{2n}$ (of order $2n$) is $\operatorname{Aff}(\mathbb Z/n\mathbb Z)$, the group of affine transformations of the $\mathbb ...
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Proving the Well-Ordering Property

My textbook states the Well-Ordering property as following: If $A$ is any nonempty subset of $\mathbb{Z}^+$, there is some element $m \in A$ such that $m \le a$, for all $a$ in $A$ ($m$ is called ...
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83 views

Prime ideals in formal power series

Let $A$ be a commutative ring with unit. If $\mathfrak{p} \subset A $ is a prime ideal, then $\mathfrak{p}$ is the contraction of a prime ideal of $A[[x]]$, the ring of formal power series. Why is ...
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The pre-image of an irreducible element.

Problem: Let $ R,S $ be integral domains and $ f: R \to S $ a unit-preserving homomorphism. Assume that $ x \in S $ is irreducible. Then does the pre-image $ {f^{-1}}[\{ x \}] $ contain only ...
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Understanding Newton polygons

I'm trying to understand the concept of the Newton polygon in relation to factoring polynomials over a field. I understand given a polynomial $$ 1+a_1x + a_2x^2+\cdots+ a_nx^n$$ that the Newton ...
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64 views

Characterize elements in $\mathbb Q(\sqrt2,\sqrt[3]5)$

I'm studying algebra by Herstein's Topics in Algebra, 2ed, and stuck at Ex.5.1.6(b): In $\mathbb Q(\sqrt2,\sqrt[3]5)$ characterize all the elements $w$ such that $\mathbb Q(w)\ne \mathbb ...
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Classification of separable algebras up to Morita equivalence

Is there a simple classification of separable algebras up to Morita equivalence, working over a particular field $k$? For example, over $\mathbb{C}$, every separable algebra is Morita equivalent to ...
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Let $U_g$ be the group of units of $\mathbb{Z}/g\mathbb{Z}$ . . .

Let $U_g$ be the group of units of $\mathbb{Z}/g\mathbb{Z}$. Then $U_g$ is a subgroup of itself. For every unit $c$ of $U_g$, show the coset, $cU_g = U_g$. Show that the product of the elements of ...
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A question about Lang's explanation of ordered fields on pg 449

Let $K$ be a field, and $P$ the set of positive elements. We know that $P$ is closed under addition and multiplication. It is also easily seen that $1\in P$. Assume that $x\in P$. Then $xx^{-1}=1$. ...
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Number of elements and units in quotients of $\mathbb Z[i]$

I was given this problem but have no idea how to solve it. How many elements are there in $\Bbb Z[i]/(3)$? In $\left(\Bbb Z[i]\right)/(3+2i)$? In $\Bbb Z[i]/(5)$? How many units are there in each of ...
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Matrix restoring (modulo n)

Let $m,n\geqslant 2$, and $A\in \mathcal{M}_n(\mathbb Z)$ such that $\det A \equiv 1 \pmod m$. Does it (necessarily) exist $M\in \mathrm{GL}_n(\mathbb Z)$ such that $A\equiv M \pmod m$?
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The largest integer less than $n$ is $n-1$

Let $n$ be a positive integer. Prove that the largest integer which is less than $n$ is $n-1$. Attempt of a solution: Since $n$ is a positive integer $n>0$. I think I have to use the well-ordering ...
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How to count the number of elements of given order?

I am trying to prove the following result. Let $G$ and $G'$ be two finite abelian groups. Besides, they have the same number of elements of any given order. Prove that $G\cong G'$. My attempt is ...
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Showing that $U(2^n)$ is not cyclic for $n \ge 3$

I'd appreciate a hint on how to solve the following problem: Prove that $U(2^n) (n \ge 3)$ is not cyclic. ($U(m)$ is the group of positive integers $j \le m$ such that $\gcd(j,m)=1$, under ...
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Showing that $\mathbb{R}\times [0,1]$ is quasi-isometric to $\mathbb{R}$

I have a question that asks me to show that $\mathbb{R}\times [0,1]$ is quasi-isometric to $\mathbb{R}$ I have having trouble showing what I have is a quasi isometry. My map is simply: ...
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How many Groups there are on a finite set?

Let say cardinality of set S is $n=|S|$. We know that there are $n^{n^2}$ all binary operations on that set. To find out how many groups can be created by this set and by those operations, we need not ...
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Action of factor group on a group

Suppose $A$ and $G$ are finite groups and $A$ acts on $G$, written $g^a$ for $g\in G$, $a\in A$. If $N\unlhd A$, does $A/N$ then act on $G$ also? By $g^{[a]} = g^a$? Or do I need to assume something ...
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Need help with this exercise about real division algebra

I am trying to solve the following exrcise: Prove that complex multiplication does not extend to a multiplication on $\mathbb R^3$ so as to make $\mathbb R^3$ into a real division algebra. I ...
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finite dimensional integral domain containg $\mathbb C$

Let $ R$ be an integral domain containing $\mathbb C$. Suppose that $R$ is a finite dimensional $\mathbb C$-vector space . Show that $R=\mathbb C$. One side $\mathbb C \subset R$ is obvious. What ...
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I need an example of a function with these properties.

I have a problem that says $A$ is finite and $B\subset A$ and that $G$ is the subset set of $S_A$ consisting of all the permutations $f$ of $A$ s.t. $f(x)\in B \ \forall \ x\in B$. these functions ...
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there is no injective group homomorphism from $\mathbb Z\times\mathbb Z$ into $\mathbb Z$

there is no injective group homomorphism from $\mathbb Z\times\mathbb Z$ into $\mathbb Z$ But i don't know why it is true. should i investigate all group homomorphisms from $\mathbb Z\times\mathbb ...
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Involutions of the second type in a division algebra

I'm trying to figure out some details about involutions of division algebra, thought maybe someone here might have a better insight. Let $k$ be a $p$-adic or number field, and let ...