# 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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### Exactness of localization functor for noncommutative rings, and categories with left/right calculi of fractions

I know that if $\mathsf C$ is a category and $\Sigma$ is some class with a right calculus of fractions then the localization functor $\varphi: \mathsf C\longrightarrow \mathsf C[\Sigma ^-1]$ is right ...
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### What can we say about the dual of an $R$-module homomorphism?

Suppose $R$ is some ring (not necessarily commutative) and let $M,\,N$ be $R$-modules. Now let $f:M\to N$ be an $R$-module homomorphism. If ${}^\ast$ denotes duality, then we can also consider the ...
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### Subring generated by idempotents

Assume that $R$ is a simple ring possessing an idempotent $e\not =0,1$. If $R'$ is the subring of $R$ generated by the idempotents, one could prove that $eRe⊆R'$ (and of course, that ...
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### euclidean division for polynomials with coefficients in noncommutative rings

I know that if $R$ is a commutative unitary ring, $f(x), g(x) \in R[x]$, and the leading coefficient of $g$ is a unit in $R$, it is possible to divide $f$ by $g$. My question is: is commutativity of ...
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Let $A \subset k[x_1, \dots, x_n]$ be a subalgebra, which is also a graded subspace$$A = \bigoplus_{i \ge 0} A_i.$$One can write$$A = A_0 \oplus A_{> 0},$$where we have$$A_0 = k^0[x_1, \dots, x_n] ... 0answers 47 views ### Show that \text{deg}(M_n(\mathbb{K})) = n, where \mathbb{K} is a field. Definition: Let A be a ring and Z=Z(A) its center. We say that t \in A is algebraic over Z if there exist z_0,z_1, \ldots , z_n \in Z such that$$z_0+z_1t+ \cdots + z_n t^n = 0 \quad ...
Let $J$ be an ideal of $R$, and $M$ a right $R$-module. Since $Jr \subseteq J$, $M / MJ$ is naturally a right $R$-module. Since it seems relevant to another problem, I am trying to determine ...