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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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1answer
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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1answer
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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1answer
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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0answers
423 views

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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1answer
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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1answer
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 ...
1
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1answer
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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2answers
193 views

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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1answer
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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2answers
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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0answers
74 views

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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1answer
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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2answers
95 views

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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1answer
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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0answers
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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0answers
89 views

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 ...
3
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1answer
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 $Y$ ...
3
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1answer
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 ...
3
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1answer
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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0answers
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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0answers
91 views

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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1answer
119 views

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 ...
3
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1answer
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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3answers
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 ...
2
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1answer
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 ...
9
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1answer
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\...
2
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1answer
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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0answers
137 views

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 ...
5
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1answer
161 views

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 ...
11
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1answer
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 \...
3
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1answer
83 views

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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0answers
213 views

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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2answers
493 views

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 ...
1
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1answer
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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62 views

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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1answer
81 views

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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0answers
93 views

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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1answer
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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1answer
137 views

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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1answer
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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1answer
91 views

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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0answers
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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1answer
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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1answer
74 views

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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1answer
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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1answer
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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1answer
254 views

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 $\...
3
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1answer
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 ...
1
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2answers
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, $(I^2:x_1x_5-x_2x_4)=I$ ?