# Tagged Questions

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### Does this complex remain exact after I restrict the maps?

$R$ is a commutative ring with unity. Assume you have two matrices $A:R^n\rightarrow R^m$ and $B:R^m\rightarrow R^n$ such that they form an exact complex in the obvious way, i.e., \cdots\rightarrow ...
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### Hamiltonian Quaternions: A Call for Counterexample for ($AB=I_m \implies n≥m$)

How can I build a counterexample to $AB=I_m \implies n≥m$ in the ring of Hamiltonian quaternions? Notice that the vectors $(1,i)$ and $(j,k)=(1,i)j$ are linearly dependent as vectors of the right ...
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### Question about Noncommutative Rings by I. N. Herstein

Here's an example that I do not quite understand it fully, it's on page 6 of the book. And here's what it says: Let $F$ be a field, and $F_n$ be a ring of all $n\times n$ matrices over $F$. We ...
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### Finding invariant factors of finitely generated Abelian group

There is this question that I wasn't sure how to do but somehow got the answers partially correct (maybe). Suppose that the abelian group $M$ is generated by three elements $x,y,z$ subject to the ...
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### Some questions on invariant factor decomposition of modules

Let $M$ be the $\mathbb{Z}$-module $F/N$ where $F=\mathbb{Z}^2$ and $N=\langle(6,4),(4,8),(4,0)\rangle\le\mathbb{Z}^2$. Consider the homomorphism $\varphi:\mathbb{Z}^3\to\mathbb{Z}^2$ such that the ...
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### Some questions on Proof of Structure Theorem

I understand the general idea of the proof of Structure Theorem for finitely generated modules over a principal ideal domain but I found it quite difficult to follow some lines of reasoning in the ...
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### How to programmatically find the inverse of a 2x2 matrix (mod 26).

I'm trying to create a hill cipher utility. One feature I want is to be able to compute the key if you have the plaintext and ciphertext. $C$ = ciphertext matrix ($2\times 2$), $P$ = plaintext matrix ...
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### Rewriting automorphism of matrix algebra in terms of automorphisms of the underlying ring?

I've used the following idea as a black box for some time now, but it occurred to me I don't fully understand why it's true. Suppose $A=M_n(R)$ is the algebra of square matrices over some division ...
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### When do the invariant factors of a direct sum of diagonal matrices correspond to those of its summands?

I am trying to prove something about matroids, which I have reduced to the following question: Suppose I have a matrix $M$ which is a direct sum of submatrices $M_1,M_2,\ldots,M_k$. When do the ...
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### Classification of simple modules for algebra of upper triangular matrices?

I've been refreshing my linear algebra, and this is a question of curiosity I have. Let $U:=U_n(F)$ be the algebra of upper triangular $n\times n$ matrices over a field $F$. Is there a classification ...
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### Characterizing matrices over a division ring as a product of a simple module?

I was reading about semisimple modules, and there's a fact relating to simple modules that I don't recall. Suppose you have a ring $R=M_n(D)$, by which I mean the $n\times n$ ring of matrices with ...
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### On rank of a matrix whose entries are polynomials

(I took courses on linear algebra, but I don't know anything about $R$-modules or such things.) How do you define the rank of a matrix whose entries are polynomials in $K[X]$? If you assign some ...
I need to check only one axiom for matrix bimodule made from certain bimodule. Here some preparatory definitions. Matricial approach. (See here for details) For arbitrary linear space $V$ denote by ...