Tagged Questions

For questions about rings which are not necessarily commutative and modules over such rings.

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Simple modules of an algebra

How can we find the simple modules of this algebra $$\begin{pmatrix} k & 0 &0 \\ k & k & 0 \\ k&0&k \end{pmatrix}$$ And why this algebra is not semisimple(i,e it is ...
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Aside from Matrix Multiplication, when else is multiplication not commutative?

Nearly all of my experience with math is in the "applied math" realm, so I haven't had any formal study of rings, or other fundamental algebraic concepts that help to prove all the relevant applied ...
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Understanding dual (noncommutative) modules / vector spaces functorially

For vector spaces over a fixed field $\Bbbk$, there's a natural transformation from the identity functor to the double dual functor. I think here's a way to construct it. Start from the identity arrow ...
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Show $R$ is right-Artinian but not left-Artinian

Let $R$ be the subring of $M_2(\mathbb{R})$ defined by $$R=\left\{\begin{pmatrix}a&b\\0&d\end{pmatrix}\mid a\in\mathbb{Q},b,d\in\mathbb{R}\right\}$$ Show that $R$ is right-Artinian but not ...
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Proof for a ring being right Artinian but not left Artinian

I am currently looking at the following example (and other similar examples) and I can follow the proof that it is a right Artinian ring and I also follow the example given as to why it is not a left ...
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Version of Wedderburn's theorem on central simple algebras

Suppose that $A$ be a central simple algebra over a field $k$. Then by Wedderburn's theorem $A\cong M_n(D)$ for some division $k$-algbera $D$. But to define the 'Brauer equivalence' I need that $D$ is ...
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Non-abelian groups of order less than or equal to 150

Please is there anywhere one could see a classification of nonabelian groups of orders less than or equal to 150?
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Showing a $k$-algebra isomorphism

Suppose $A$ is a finite dimensional $k$-algebra. I want to show that $A\otimes_{k}M_n(k)\cong M_n(A)$ for any positive integr $n$. Can you show how to proceed?
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Solving non-commutative “quadratic” equation with inhomogenously typed coefficients

Is there a general method to solve for $z\in\mathbb{R}^d$ in the non-commutative $z^\intercal\alpha z + \beta z + \gamma = 0$ where $\alpha\in\mathcal{M}_d(\mathbb{R})$ (real $d\times d$-matrix), ...
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Tensor Product of irreducible modules

Let $A$ be a $\mathbb C$ algebra. Let $S$ be an irreducible $A$ module? Then what $S \otimes_A Hom_A(S,S)$? Is it equal to S? I know that $S \otimes_{\mathbb C} Hom_A(S,S)$ is isomorphic to $S$ as a ...
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Noncommutative free resolution

This is probably something very simple but I got stucked with it. Consider the polynomial ring $k[x_1,\dots,x_n]$. How can one construct a free resolution of it in the category of $k$-algebras(not ...
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A monoid where every element has finitely many divisors

Is there a special name or has there been any study of monoids of this form? This came up in considering the general construction of a multivariate power series algebra over a ring $R$; usually we ...
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Isomorphism of Submodules extends to Automorphism

In [Lam, Lectures on Modules and Rings] there is the following proposition (15.20): Let $M$ be finitely generated projective over the Quasi Frobenius ring $R$, and $A,B$ be submodules of $M$. Then ...
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Flatness via factoring homomorphisms

Theorem (4.32) of "Lectures on modules and rings" by T.Y. Lam says that a module $P_R$ is flat iff any $R$-homomorphism $λ:M→P$ where $M$ is any finitely presented $R$-module can be factord ...
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What do we call the ring homomorphism $R \rightarrow \mathrm{End}_{\mathbf{Ab}}(X)$ associated with an $R$-module $X$?

First, a convention: given an abelian group $X$, write $\mathrm{End}_{\mathbf{Ab}}(X)$ for the set of all group homomorphisms $$X \rightarrow X.$$ Now let $R$ denote a ring. Question. Given an ...
Let $M$ be an indecomposable injective right module over a right Artinian ring $R$, so $M$ has exactly one associated prime ideal $P$ (Lectures on Modules and Rings, T.Y. Lam). Now, $R/P$ is a simple ...