# 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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92 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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### 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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### 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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246 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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163 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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81 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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85 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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471 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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187 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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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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464 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 ...