Questions tagged [abstract-algebra]

For questions about groups, rings, fields, vector spaces, modules and other algebraic objects. Associate with related tags like group-theory, ring-theory, modules, etc. to clarify which topic of abstract algebra is most related to your question and help other users when searching.

57,551 questions
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Isomorphism problem for the center of modular group algebras

Let $p$ be a prime number, $G,H$ a finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $1+\operatorname{rad}(KG)$ is a p-group containing $G$. My ...
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Prove that these two conditions are equivalent

Let $m, n \in \mathbb{N}$. Prove that these two conditions are equivalent: $m, n$ are coprime; for any group $G$, for any subgroup $A \subseteq G$ of order $m$ and for any subgroup $B \subseteq G$ of ...
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If $R$ is a PID then $\mathrm{Spec}(R)\!=\!\mathrm{Max}(R)\!\cup\!\{0\}$? [on hold]

Is the following statement true? If $R$ is a PID then $\mathrm{Spec}(R)\!=\!\mathrm{Max}(R)\!\cup\!\{0\}$ I know the reverse is false.
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Do we lose much if we take isomorphism as equality in a purely algebraic context?

In algebra, we sometimes end up with a sentence relating two objets via an isomorphism. As algebra is kind of spiritual behavior of objects, (we almost never speak about the object, but we speak about ...
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Circle group $S^1$

Can someone describe the circle group $S^1$ in a easily understandable way? Are elements in $S^1$ in the form of $e^{2i\theta\pi}$? What does this look like pictorially? Doesn't necessarily need a ...
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In general, what's the relation between $\text{Ext}^n_R(A,B)$ and $\text{Ext}^n_R(B,A)$ (if any)?

Here $A,B$ are arbitrary $R$-modules. Similarly, what's the relation between $\text{Tor}^n_R(A,B)$ and $\text{Tor}^n_R(B,A)$? I am on 17.1 Dummit and Foote and couldn't find any remarks on this.
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Does the category of $(R,S)$-bimodules contain a free object on one generator?

I believe that in the category of $(R,S)$-bimodules where $R$ and $S$ are rings with identity, then ${}_R R \otimes_\mathbb{Z} S_S$ is a free object on one generator in this category. But, if $R, S$ ...
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Identifying simple tensors.

Let $S$ be a domain. I want to determine whether or not, every element of $\text{Frac(S)}\otimes_S M$ is a simple tensor, where $M$ is any $S$-module. I couldn't produce a tensor that is not pure in ...
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Is $X\cong\textrm{Ker}(f)\oplus\textrm{Im}(f)$ for a module homomorphism $f:X\to Y$ with semisimple domain?

Let $f:X\to Y$ be a module homomorphism with semisimple domain. Does \begin{equation*} X\cong\textrm{Ker}(f)\oplus\textrm{Im}(f) \end{equation*} hold true that? (In my previous question it was kindly ...
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Mapping from $\mathbb Z/2\mathbb Z \times \mathbb Z/3\mathbb Z \times \dots\times \mathbb Z/p_n\mathbb Z$ to $\mathbb Z/p_n\#\mathbb Z$.

I know $\mathbb Z/2\mathbb Z \times \mathbb Z/3\mathbb Z \times \dots\times \mathbb Z/p_n\mathbb Z$ is isomorphic $\mathbb Z/p_n\#\mathbb Z$ (where $p_n\#$ is the primorial of primes up to $p_n$) by ...
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