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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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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 $x,y\... 2answers 667 views ### Cardinality of certain subsets in vector spaces over finite fields Assume that you have an$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$... 1answer 179 views ### Interpretations for higher Tor functors Let's work in the category$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$(Yoneda ... 1answer 93 views ### Do these sorts of submonoids go by a particular name? Given any monoid$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?... 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{...
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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 ...
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 ...
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 ...
255 views

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 $$[T_1^{n_1},T_2^{n_2}]=\displaystyle\sum_{\substack{\alpha+\alpha'=n_1-1 \\ \beta +\beta'=n_2-... 1answer 175 views ### Localization of the injective hull Let R be a Noetherian commutative ring. Let E(M) denote the injective hull of M. I want to show that E(M)_\mathfrak{p}\simeq E(M_\mathfrak{p}) for any \mathfrak{p}\in \text{Spec}(R). To do ... 1answer 84 views ### Generalizing a codistributive property of sufficiently disjoint normal subgroups to protomodular categories In a poset, whenever the meets and joins below exist, their universal properties induce a containment$$(A\vee B)\wedge (A\vee C)\geq A\vee(B\wedge C). This is an instance of codistributivity. In a ...
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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 ...
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 ...
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 ...
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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