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        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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        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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        votes
        1answer
        186 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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        votes
        1answer
        153 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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        399 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 ...
        1
        vote
        1answer
        240 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-...
        3
        votes
        1answer
        149 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
        76 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
        188 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 ...
        1
        vote
        1answer
        83 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
        415 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 ...
        3
        votes
        0answers
        68 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, ...
        3
        votes
        1answer
        176 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
        94 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 ...
        10
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        1answer
        428 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 ...
        1
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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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