# Tagged Questions

Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, among other topics.

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### Are “noncommutative resolutions” a thing?

I am interested in looking at "resolutions" of modules in noncommutative rings, making some obvious necessary modifications to the definition of a resolution. However, whenever I try to search the ...
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### Why is the product of two negative numbers not negative? [duplicate]

A positive number multiplied by a positive number is positive, but negative multiplied by negative is not negative but it becomes positive. How ?
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### Examples of groups that are not subgroup [on hold]

Sorry, my English is poor, but I have a question. It is usual to find the definition of subgroup as: "We define a subgroup $H$ of a group $G$ to be a nonempty subset $H$ of $G$ such that when the ...
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### $\mathbb{F}$-subalgebra generated by a set

Assume that $A$ is an $\mathbb{F}$-algebra, where by $\mathbb{F}$ I just denote an arbitrary field. Furthermore, if $X \subset A$ is a proper subset, how do we define the $\mathbb{F}$-subalgebra of $A$...
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### Using the Tensor Product Construction to Show Linear Independence of the Standard Basis

For simplicity's sake, let's assume $V = \mathbb{R}^2$ with standard basis $e_1, e_2$. I construct the tensor product $V \otimes V$ as the quotient of the the free vector space $F(V \times V)$ by the ...
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### Ideal and subring of $End_K(V)$

Let $V$ be an infinite-dimensional $K$-vector space, $E = End_K(V)$ and $I = \{f \in E;\,\,dim(f(V)) < \infty\}$. I want to prove that $I$ is an ideal of $E$ and $Kid_V + I$ is a subring of $E$. ...
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### Why would I learn modern category theory if my interest mainly is structured sets, what would I have to gain? [closed]

A long time ago I studied mathematics at the University of Stockholm. I had a romantic view of modern algebra and manage to make the first two algebra courses by self studies in order to immediately ...
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### Confusion about generated subrings and subfields

In Milne's field theory notes he defines, given an extension $E/F$ and a subset $S\subset E$, the subfield of $E$ generated by $F$ and $S$ as the smallest subfield of $E$ containing both $F$ and $S$. ...
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### Advantages and disadvantages of a particular definition of rings and subrings

My professor defines a ring as a triple $(A, +,\ \cdot)$ such that $(A, +, 0)$ is an abelian group, $(A,\ \cdot)$ is a semigroup and $\cdot$ distributes over $+$. Subsequently, $B\subseteq A$ is said ...
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### Does the uniqueness of splitting fields up to non-canonical isomorphism fall from algebraic geometry?

Does the uniqueness of splitting fields up to isomorphism over the base field fall out from some deep results in algebraic geometry, or is it something special to fields which I shouldn't expect to ...
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### $f:A\to B$ flat homomorphism of rings, $Q$ is a prime ideal of $B,$ and $P=Q^c.$ Why is $B_Q$ a local ring of $B_P?$

I have a question about Exercise 2.18 on Introduction to Commutative Algebra by Atiyah and MacDonald. Let $f:A\to B$ be a flat homomorphism of rings, let $Q$ be a prime ideal of $B$ and let $P=Q^c,$ ...
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### $F(a_1,\ldots,a_n) = F(a_1+t_2 a_n+\cdots+t_n a_n)$ if Extension is Simple?

Suppose that F is an infinite field and that $L = F(a_1,\ldots,a_n)$ is a simple extension of $F$. Can you show, without the assumption that $[L:F] < \infty$ that $\exists t_2,\ldots,t_n \in F$ ...
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### S is a subgroup of S(n) such that X(n)=n,X belongs to S(n). then show that S is isomorphic to S(n-1). [closed]

Pls help me with define a isomorphic function from S to s(n-1)
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### If $G$ is a finitely presented group then is the commutator of $G$ isomorphic to the commutator of $F$ mod the relations?

Let $G=\langle\ S\ |\ R\ \rangle$ be a finitely presented group. Let $F$ be the free group with generating set $S$. Let $[F,F]$ and $[G,G]$ be the commutator subgroups of $F$ and $G$ respectively. Let ...
### find a 2 sylow subgroup of $G=S_4\times S_3$ and a $3$ sylow subgroup. [closed]
how to find a $2$ sylow subgroup of $G=S_4\times S_3$ and a $3$ sylow subgroup. and the number of each number of subgroup.