# Questions tagged [abstract-algebra]

Deprecated; do NOT use this tag. Instead you could consider gr.group-theory, ac.commutative-algebra, ra.rings-and-algebras, universal-algebra, or various more specific tags.

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### Do these sorts of submonoids go by a particular name?

Given any monoid $M$ for every element $x\in M$ we can define two submonoids of $M$ as follows:
$$r(x)=\{y\in M:xy=x\}$$
$$l(x)=\{y\in M:yx=x\}$$
Do these sorts of sub-monoids go by a particular name?...

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78 views

### How to show the set $\operatorname{Hom}_K(L,\bar{K})$ of all $K$-embeddings of $L$ is partitioned into $m$ equivalence classes of $d$ elements each? [closed]

Let $L|K$ be a finite separable extension. Denote the algebraic closure of $K$ by $\bar K$.
$\forall x\in L$, denote $d=[L:K(x)]$ and $m=[K(x):K]$.
How to show the set $\operatorname{Hom}_K(L,\bar{...

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195 views

### Origin of the concept of “homomorphism”? [duplicate]

When was the concept of a "homomorphism" of algebraic structures first introduced?
Steinitz' 1910 paper Algebraic Theory of Fields is often pointed to as the first true work of abstract algebra, yet ...

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161 views

### Alternate descriptions of finite fields

The finite field of order $p^n$ is isomorphic to $(\mathbb Z/p \mathbb Z)[X]/(P)$, where $P$ is an irreducible polynomial in $(\mathbb Z/p \mathbb Z)[X]$ of degree $n$. This describes every finite ...

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### Why does $E\otimes_KE\cong EG$ imply that Galois theory works?

This is a part of statement in the book I do not fully appreciate. Suppose $E/K$ is Galois extension and $G$ the galois group of $E/K$. $E[G]$ is the group ring formed by finite group $G$.
"It is ...

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244 views

### Commutator of the power of two elements [closed]

Let $T_1,T_2\in \cal{A}$ with $\cal{A}$ is an algebra.
Let $n_1,n_2\in \mathbb{N}$. Is it true that
$$[T_1^{n_1},T_2^{n_2}]=\displaystyle\sum_{\substack{\alpha+\alpha'=n_1-1 \\ \beta +\beta'=n_2-...

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155 views

### Localization of the injective hull

Let $R$ be a Noetherian commutative ring. Let $E(M)$ denote the injective hull of $M$. I want to show that $E(M)_\mathfrak{p}\simeq E(M_\mathfrak{p})$ for any $\mathfrak{p}\in \text{Spec}(R)$.
To do ...

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79 views

### Generalizing a codistributive property of sufficiently disjoint normal subgroups to protomodular categories

In a poset, whenever the meets and joins below exist, their universal properties induce a containment $$(A\vee B)\wedge (A\vee C)\geq A\vee(B\wedge C).$$ This is an instance of codistributivity. In a ...

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### Is the triple product in a Freudenthal Triple System fully symmetric?

I'm trying to learn about Freudenthal Triple Systems. Here is the definition given by Helenius [1], start of Section 5:
A Freudenthal triple system is a finite-dimensional vector space $V$
over a ...

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84 views

### When an ideal is locally comaximal with idempotents(restated)

I saw the following question at mathstackexchang < https://math.stackexchange.com/questions/2282194/when-an-ideal-is-locally-comaximal-with-idempotents>. It seems to be a nice question and I need ...

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455 views

### Direct sum of injective modules is injective

By the Bass-Papp Theorem, for a unital ring $R$, any direct sum of injective left $R$-modules is injective if and only if $R$ is left Noetherian. I would like to restrict my consideration to an ...

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### The group of $k$-automorphisms of $k[x_1,\ldots,x_n,x_1^{-1}]$.

Let $k$ be a field (of characteristic zero).
For $k[x_1,\ldots,x_n]$ it is known that the affine and triangular automorphisms generate $G_n$, the group of automorphisms of $k[x_1,\ldots,x_n]$,
see, ...

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181 views

### Short proof a monoid is a group iff every splitting is right homogeneous

In the paper "Schreier split epimorphisms between monoids" by Bourn, Nelson, Martins-Ferreira, Montoli and Sobral, Semigroup Forum
June 2014, the authors prove a characterization of groups among ...

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### Powers of small square matrices over the Laurent polynomial ring with integer coefficients

I'm trying to calculate the powers of a 2 by 2 matrix with entries in $\mathbb{Z} \left[ t,t^{-1} \right]$.
The matrix is \begin{bmatrix}
0 & 1 \\
1 & t
\end{bmatrix}
I tought of writing my ...

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447 views

### Do there exist nonzero identically vanishing polynomials over infinite (or characteristic zero) reduced indecomposable commutative rings?

Let $R$ be an infinite, characteristic zero, commutative ring. I can furthermore suppose it is reduced and indecomposable (no nontrivial nilpotents or idempotents).
My question is whether there is a ...

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50 views

### Any link between abelian $R/J(R)$ and 2-primal condition

Let $R$ be noncommutative unital ring such that each element of the quotient $R/Soc(R_R)$ is idempotent. If the nilpotent elements of $R$ form an ideal, is it true that the idempotents of $R/J(R)$ ...

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### A Boolean quotient ring of a prime ring

I am searching for a unital prime ring $R$ such that its right socle $Soc(R_R)$ is nonzero and proper, and such that $R/Soc(R_R)$ is a Boolean ring (i.e., all its elements are idempotent).
Thanks for ...

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303 views

### Regular functions on a product of varieties

Let $k$ be an algebraically closed field and let $X$, $Y$ be varieties over $k$.
Let us denote by $\mathcal{O}(X)$ and $\mathcal{O}(Y)$ the $k$-algebra of regular functions on $X$ and $Y$ ...

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181 views

### Quasi thin Groups and classification theorem

I read the following paragraph from Serre's book (Topics in Galois Theory).
Although the proof of the classification theorem has been announced, described, and advertised since 1980, it is not yet ...

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212 views

### What is the relation between cobar duality and Feynman transform

If $O$ is a cyclic operad, it can be regared as a modular operad $P$ with $P(g,n)=0$, for $g >0$. So we have cobar dual $BO$ and Feynman transform $FP$(with trivial cocycle). Is there any ...

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297 views

### Does Alexander-Whitney formula imply Pythagoras theorem? [closed]

There are many diverse proofs of the Pythagorean theorem, which says something non-trivial about the diagonal of the standard square. Its length may be approximated by the convergents $1, \frac{3}{2}...

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### Monomorphism between two ideals

Let $I $ and $J $ be two ideals in a commutative ring $R $ with $1$. Is there any equivalent property for the fact that there are no $R $-module monomorphism from $I $ to $J $?

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### Algebraization of Bayesian networks?

The algebraization of classical propositional logic is Boolean algebra.
Bayesian networks are a generalization of classical propositional logic with probability truth-values.
What is the ...

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311 views

### Ternary associative multiplication

In this answer Brian M. Scott describes the following generalization of a binary associative multiplication to a ternary one: it is a function $$[\cdot,\cdot,\cdot] : G\times G \times G \to G$$ such ...

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344 views

### Non-free projective pearls (general and Abelian)

A pearl is an ordered pair $\ \mathbf P:=(G\,\ S),\ $ where $\ G\ $ is a group, and $\ S\ $ is a non-empty subset of G which does not contain the neutral element of $\ G\ $ (i.e. not 1 in the ...

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103 views

### Right socle of a group ring

Let $p$ be a prime number and $n$ a positive integer. I want to know what is the (right) socle of the group ring $A=\mathbb Z_{(p)}C_n$, where $\mathbb Z_{(p)}$ is the localization of integers at the ...

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204 views

### Maximum cardinal of a set of linearly independent vectors in a module

A student asked me this, and I can't believe I never knew the answer to this.
Let $R$ be a commutative ring, and $M$ be an $R$-module.
If $M$ has a set of $n$ linearly independent vector for each $n\...

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304 views

### How a group cocycle becomes a group coboundary in a smaller group

Let $A$ be a group. Then we choose that $B$ is a subgroup of $A$.
Let us write the cohomology group cocycle of $A$, as $\alpha_{d}(\{a\}) \in H^d(A,U(1))$ where $\{a\}$ is a shorthand for a set of $...

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### Can completely multiplicative functions be extended to $\overline{\mathbb{Q}}$ or further?

I'm looking for a subject of study that handles the following question. I'm not the most familiar with abstract algebra; I have a strong working knowledge and that's about it, but I've been ...

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### Coherent subsheaf of co-admissible modules of Schneider and Teitelbaum

Let $M$ be a co-admissible module over a Frechet Stein Algebra $A=\varprojlim A_{q_n}$ as in this paper. Let $N$ be a closed submodule of $M$. I have some difficulty in understanding lemma $3.6$ of ...

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311 views

### Torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z^2$

Is there a torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z \oplus \Bbb Z$ (as groups)?
Notice that $\Bbb Z$ is not cancellable, so
$A \oplus \Bbb Z \...

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### Isomorphism concerning $Soc(M_n(R))$

It is known that $M_n(R/J(R))\simeq M_n(R)/M_n(J(R))=M_n(R)/J(M_n(R))$. I tried to prove the same "isomorphism" replacing $J(R)$ by $Soc(R_R)$, where $J(R)$ and $Soc(R_R)$ stand for the Jacobson ...

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### Does the tensor algebra $T(V)$ of $V$ isomorphic to the symmetric algebra of the free Lie algebra over $V$?

Let $V$ be a finite dimensional vector space. Let $T(V)$ be the tensor algebra over $V$.
Do we have $T(V) \cong S(Lie(V))$ as a graded vector space? Here $S(Lie(V))$ is the symmetric algebra of the ...

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### Has the Jacobson/ Baer radical of a group been studied?

On groupprops, the Jacobson or Baer radical of a group $G$ is defined to be the intersection of all maximal normal subgroups of $G$. This is similar to, but distinct from, the Frattini subgroup which ...

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215 views

### Galois group of an L-function

Let $ M $ be a class of L-functions such that whenever $ F $ and $ G $ belong to $ M $, then so do their product $ F.G $ and their tensor product $ F\otimes G $ defined by $ F\otimes G : s\...

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### Homomorphic image of $B_{\lambda}^o(S)$ is the Brandt $\lambda^o-$ extension of some monoid with zero

Let $S$ be a monoid with zero and $I_{\lambda}$ be an indexed set, then $B_{\lambda}(S) = \{ (\alpha, s , \beta ) : \alpha , \beta \in I_{\lambda}, s\in S \} \cup \{0\}$ is a semigroup and $J = \{ (\...

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### Endomorphism of Brandt Semigroup $B_n(G)$, where $G$ is a finite group

I want to show that $End_0 (B_n(G)) = \cup\phi_{\sigma,g} \cup C_{I(B_n(G))}$, where $\phi_{\sigma,g} : B_n(G) \rightarrow B_n(G) $ is an endomorphism is defined by $(i,a,j)\phi_{\sigma,g} = (i\sigma ...

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### Preimage of projection of idèles, and other usual maps

Let $K$ be a quadratic number field.
I am struggling with some "usual" maps in algebraic number theory, but with which I am not used to, confusing a lot of different settings, as idèles, ...

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403 views

### What's the cokernel of a monoid homomorphism?

Let $f:A\to B$ be a monoid homomorphism. Where can I find an explicit description of the its cokernel? Are there any books on this topic?
If anyone cares, here's my motivation. In the category of ...

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### Trying to understand the proof of Laurent phenomenon of cluster algebras

I am trying to understand the proof of Laurent phenomenon of cluster algebras in the book (Sergey Fomin, Lauren Williams, Andrei Zelevinsky, Introduction to Cluster Algebras. Chapters 1-3, arXiv:1608....

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352 views

### Is there a theory of decomposition into indecomposables? What's the relation to idempotents?

Call a nonzero object of a pointed category simple if it has no proper quotients, and indecomposable if it's not the product of two objects (dual to connected).
Idempotents seem to pop up in many ...

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### Do we have a one to one correspondence between positive roots and reflections in a Coxeter group?

By the answer of the question, the set of reflections of a Coxeter system $(W,S)$ is given by $R = \{ wsw^{-1} : w \in W, s \in S\}$.
Do we have a one to one correspondence between positive roots and ...

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148 views

### Relationship between pointed protomodularity and $\operatorname{Coker}\ker f\overset{\cong}{\longrightarrow}\operatorname{Ker}\operatorname{coker}f$

In hopes of understanding algebra better, I've been reading here and there about protomodular categories and the like. Among other things, the theory surrounding these (and some other) categories ...

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175 views

### What are all primitive elements in a tensor algebra?

Let $H$ be a Hopf algebra and $V$ a Yetter-Drinfeld module over $H$. Then there is a braiding $\Psi: V \otimes V \to V \otimes V$ given by $\Psi(x \otimes y) = x_{(-1)}.y \otimes x_{(0)}$, where $x_{(-...

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### Are braided commutators primitive elements of a braided Hopf algebra?

Let $H$ be a braided Hopf algebra. The multiplication on $H \otimes H$ is defined by $(a \otimes b)(c \otimes d) = a \Psi(b \otimes c) d$, $a,b,c,d \in H$.
Let $H = T(V)$. There is a algebra map $\...

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181 views

### Is there any algorithm to find the minimal generating set for $A_{n}$

I know that finite simple groups can be generated by two elements.(See this question on MO)
So as a specific example, Take Alternating group $A_{n}$, $n>4$.
We also know that $A_{n}$ is $(2,3)$ ...

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90 views

### Free algebras on sets of different cardinality - for what theories are they non-isomorphic?

Following the case of groups, I asked in this MSE question for a quick proof that given a free-forgetful adjunction $F\dashv U$ for some algebraic theory, we have $X\not\cong Y\implies FX\not\cong FY$....

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### Primitive elements in group hopf algebras over fields of non-zero characteristic

An element $x$ of a Hopf algebra $H$, is called a primitive element if $\Delta(x)=1\otimes x+x\otimes 1$. The set of primitive elements of $H$ is denoted $P(H)$. It can be shown that:
"If $H$ is a $\...

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138 views

### Extension field $\mathbb{C}(t,u)$ over $\mathbb{C}(t^n,u^n)$

I'm teaching myself some mathematics, so post question here sometimes is my last resort to get an answer, i have already posted this question on Mathematics Stack Exchange But no one answers, and I ...

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431 views

### Properties of colon ideal [closed]

Let $R=k[x_1,\ldots,x_6]$ be a polynomial ring and $I=(x_1x_5-x_2x_4,x_2x_6-x_3x_5)$ be an ideal.
How to show that, $(I^2:x_1x_5-x_2x_4)=I$ ?