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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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        The “semi-symmetric” algebra of a vector space

        If $V$ is a vector space over a field $K$, then the symmetric algebra $S(V)$ is defined as the tensor algebra $T(V)$ factorized by the two-sided ideal generated by $x\otimes y-y\otimes x$, with $x,y\...
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        votes
        2answers
        667 views

        Cardinality of certain subsets in vector spaces over finite fields

        Assume that you have an $n$-dimensional vector space over a finite field (therefore the number of elements in the vector space is finite) and $F$ is a subset of this vector space which contains $m$ ...
        17
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        1answer
        179 views

        Interpretations for higher Tor functors

        Let's work in the category $R$-${\sf mod}$, for concreteness. I know that one can see the modules ${\rm Ext}^n_R(M,N)$ as modules of equivalence classes of $n$-extensions of $M$ by $N$ (Yoneda ...
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        1answer
        93 views

        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
        106 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{...
        9
        votes
        1answer
        216 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 ...
        2
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        1answer
        174 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 ...
        13
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        485 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
        255 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
        175 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 ...
        2
        votes
        1answer
        84 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 ...
        6
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        2answers
        204 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
        87 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 ...
        6
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        2answers
        510 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
        82 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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