# Tagged Questions

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### A matrix lies in a subring isomorphic to $\mathbb{C}$

Problem: Consider the matrix $$A = \begin{pmatrix} 0 & 3\\ -4 & 1 \end{pmatrix}.$$ Show that $A$ lies in a subring of Mat$_{2\times 2}(\mathbb{R})$ that is isomorphic to $\mathbb{C}$. ...
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### Space of matrices that commute with a given matrix

Let $A$ be an $n\times n$ complex matrix, and $C(A)$ be the vector space of all matrices that commute with $A$. I have to determinate if the dimension of $C(A)$ is greater or equal than $n$, or not. ...
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### Counting diagonalizable matrices in $\mathcal{M}_{n}(\mathbb{Z}/p\mathbb{Z})$

How many diagonalizable matrices are there in $\mathcal{M}_{n}(\mathbb{Z}/p\mathbb{Z})$ ? Where $p$ is a prime number. Attempt : By definition a matrix is called diagonalizable if there exists an ...
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### Question concerning a list sorting problem

I have the following question: Let $a,b,c,d$ be four natural numbers with $a \leq b$ and $c\leq d$. I have written a program that produces a list, which has as entries all 2-tuples $(x,y)$ with ...
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### Showing a matrix is nilpotent if its charateristic polynomial is $t^n$ mod ${\rm nil}(R)$

Let $R$ be a commutative ring. How to prove the following: If $\chi_A(t) \equiv t^n \bmod\operatorname{nil}(R)$ then $A \in M_n(R)$ is nilpotent. Note $\chi_A$ is the characteristic polynomial ...
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### Can polynomials with degree at least 2 over $\mathbb{R}$ have finite number of solutions in $End(\mathbb{R},+)$

Consider a polynomial of degree at least 2 with all coefficients in $\mathbb{R}$. We are concern with set of solution for the polynomial in $End(\mathbb{R},+)$ - the endomorphism ring of the abelian ...
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### Ring of linear transformations modulo finite rank transformations [closed]

Let $K$ be a field and $V$ be a vector space of countable dimension (infinite) over $K$, and let $L = L (V)$ be the vector space of $K$-linear transformations on $V$. Let $I$ be the ...
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I'm trying to show that the minimal polynomial of a linear transformation $T:V \to V$ over some field $k$ has the same irreducible factors as the characteristic polynomial of $T$. So if $m = ... 1answer 419 views ### How to prove that the inverse of a matrix is unique? The ring of matrix is not an integral domain. How to prove that the inverse is unique? 2answers 53 views ### The number of$n\times n$matrix over integer modulo$p$field with determinant equal$1$How to count the number of$n\times n$matrix over integer modulo$p$field with determinant equal$1$? I know that the number of invertible matrices is GL$(n,p)$. Have any ideas? 1answer 27 views ### Sylvester domains I'm an undergrad mathematics student and I'd like to request some books about Sylvester domains. Specifically I'd like to understand the fact that not all modules are Sylvester domains. I just proved ... 1answer 50 views ### Show:$\varphi\colon\mathbb{Z}_{mn}\to\mathbb{Z}_m\times\mathbb{Z}_n, k\mapsto (k\% m,k\% n)$is a ring isomorphism for$m$and$n$relatively prim Let$m\in\mathbb{Z}, n\in\mathbb{N}$. Then there exist unique elements$q\in\mathbb{Z}, r\in\mathbb{N}$with$0\leq r<n$and$m=qn+r$. We write$r:=m\% n$. Let$m,n\in\mathbb{N}$be relatively ... 0answers 52 views ### Isomorphism Linear Algebra I'm currently going through a proof and I've come across something I don't really understand: Next, an endomorphism of a left$A$-module$M$, over a ring$A$is an$A$- homomorphism ... 1answer 32 views ### Polynomial ring equals sum of sets. Let$\mathbb{K}[x]$the ring of polynomials in x with coefficients in$\mathbb{K}$. Let $$V_n = \left [nx^n + (n-1)x^{n-1} + \ldots + 1 \right ]$$ Show that $$\mathbb{K}[x] = ... 0answers 122 views ### Is this matrix decomposition possible? Given a 2\times2 matrix S with entries in \mathbb{Z} or \mathbb{Q} , when is it possible to write S=\frac{1}{3}(ABC+CAB+BCA) such that A+B+C=0, where A, B, C are matrices over the same ... 5answers 187 views ### how to prove e^{A \oplus B} =$$e^A$ $\otimes$ $e^B$ where A and B are matrices? (Kronecker operations)

how to prove $e^{A+B} = $$e^A$$e^B$ where A and B are matrices? The operations '+' and '*' are defined such that AI + IB = A+B, where I is the identity matrix. I suppose these are called Matrix ...
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### Kernel and image of map on unit group of finite field

This question is from a rings and modules course I'm doing: Let $p>2$ be prime and $f:\mathbb{F}_{p}^{*}\rightarrow \mathbb{F}_{p}^{*}$ be given by $x \mapsto x^\frac{p-1}{2}$. Prove that $ker(f)$ ...
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### direct products and direct sums of skew fields

If F is a skew field then are the arbitrary direct sums of F-modules isomorphic to their direct products (over the same index). I mean, if R is a division ring, and $\{M_i\}_{i\in I}$ some familly ...
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### How to construct the subring generated by a set, T?

I'm trying to find a constructive way of describing the subring generated by some subset, T, of a ring R. I think I could describe it as all finite sums of finite products of elements of T, but I ...
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### How to find all the roots in this ring?

Let $F:= \mathbb{F}_7[x]/(x^2+3x+1)$ Is it a field? Find all the roots in F of the polynom $f (Y) := Y^2+[3]_{F}Y +[1]_{F} \in F[Y]$. Attempt: It is a field, because $x^2+3x+1$ is irreducible ...
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### integral domain with a field as a subring [duplicate]

I would like to know if my solution to the following exercise is correct. Let $A$ be an integral domain (with a unit) which has a field $\mathbb K$ as a subring and such that $A$ is a ...
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### Prove that $M_{nm}(A \otimes B) \cong M_{n}(A) \otimes M_m(B)$

Prove that $M_{nm}(A \otimes B) \cong M_{n}(A) \otimes M_m(B)$ There's a similar question floating around but I was merely wondering if the result holds in the same way when we let A and B be fields, ...
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### Inversion in factor rings

I have this polynomials: $f = x^{4} + 3x^{3} + x^2 + 3 \in \mathbb{Z}_{5}[x]$, $g = x + 2 \in \mathbb{Z}_{5}[x]$ Does g + (f) have inversion in ring $(\mathbb{Z}_{5}[x]/(f),+,.)$ ? I should found ...
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### similar matrices over $\mathbb{Z}$

Let $A,B$ be $2\times 2$ matrices over $\mathbb{Z}$. Suppose $x^2+x+1$ is the characteristic polynomial for both $A,B$. Determine whether $A,B$ are similar to each other over $\mathbb{Z}$. How to ...
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### Annihilator of a Tensor

This is a question I have trouble understanding, hope you can clarify this to me. Problem: Find the annihilator of the tensor $e_1\wedge e_2+e_3\wedge e_4$ in ...
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### Equations over fields

Let $x_1,\cdots, x_n$ be distinct elements of a given field $F$ such that for any $k$, $\sum_{i=1}^n x_{i}^k = 0$. I want to show that all $x_i$'s are zero.
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### Why is (gcd(f,g)) = (f,g)?

f and g are polynomials of F[X]. I can't see why (f,g) = (gcd(f,g)) ? (the ideal that f and g are the generators, and the ideal that the gcd is the generator). gcd(f,g) = a*f+b*g , for specific a ...
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### Are all ring isomorphisms of $Mat_n(\mathbb{R})$ obtained by switching between bases

Let $\mathbb{R}$ denote the field of the real numbers. Let $U=(u_1,u_2,...,u_n)\in (\mathbb{R}^n)^n$ be a base for $\mathbb{R}^n$ (i.e. $u_1,u_2,...,u_n$ are a base for $\mathbb{R}^n$). Let ...
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### Any submodule U of V such that the module V/U is completely reducible must contain the radical?

Want to prove: Given an $R$-Module $V$ and $rad(V)$ which I define to be the intersection of all maximal submodules of $V$. I want to show that if, for some submodule $U$ of $V$, we have $V/U$ ...
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### homework: rings, matrices and polynomials

A,B are both nxn and diagonal matrices. Prove that there is a matrix X which is nxn, and polynomials p and q such that A= p(X), B= q(X) Is this true for ANY 2 matrices (we do not assume that they're ...
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### homework about homomorphisms : find all the homomorphisms

Find all the continuous homomorphisms $T:\mathbb{R} \rightarrow \mathbb{R}$ Find all the homomorphisms $T:\mathbb{C} \rightarrow \mathbb{C}$ (complex field) such that $T(x)=x$ for every $x$. ...
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### On the extension of fields

Let $F\subseteq K$ be a finite field extension and let $a_1,..., a_n$ be an $F$-basis for $K$. I want to show that the matrix $A := (tr(a_ia_j))$ is singular if and only if $tr K =0$. Any suggestion ...
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### Homework basic abstract algebra

My question is as follows: $R$ is a ring such that for all $x \in R, x^2=x$ $p$ is a prime ideal of $R$. Show that $R/p$ (R modulu p) has exactly 2 elements. What I did: $x^2=x$ $x^2-x=0$ ...
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### Existence of Algebra of anticommuting idempotents

Background and motivation: I'm wondering about the existence of an algebra which is in some ways similar to the exterior algebra, but is generated by idempotents rather than nilpotents. Let $V$ be a ...
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### Show that the closure of $\Bbb Q\cup\{i\}$ is a field

Let K be the closure of $\Bbb Q\cup\{i\}$, that is, $K$ is the set of all numbers that can be obtained by (repeatedly) adding and multiplying rational numbers and $i$, where is the complex square root ...
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### Prove that the set of all diagonal matrices is a subring of $\operatorname{Mat}_n(R)$ which is isomorphic to $R \times\dots\times R$ ($n$ factors)

Can someone tell me, is that diagonal matrices is a subring of $\operatorname{Mat}_n(R)$ which is (ring) isomorphic to $R \times · · · \times R$ (n factors) and why?.
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### When every matrix is a sum of nilpotent (idempotent or invertible) matrices ??

Let $R$ be a ring with non-zero identity. Consider the following three properties: Every $A \in M_n(R)$ is a sum of nilpotent matrices. Every $A \in M_n(R)$ is a sum of idempotent matrices. Every ...
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### Which is easier to work out: determinant or inverse?

Suppose $A\in M_n(R)$ be a $n\times n$ matrix over some ring $R$. Which of the following two tasks is easier? to work out $\det(A)$; to work out $A^{-1}$. More specifically, I want to know the ...
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### Let A and B be $n \times n$ real matrices with same minimal polynomial.

Let $A$ and $B$ be $n \times n$ real matrices with same minimal polynomial. Then (i) $A$ is similar to $B$. (ii) $A-B$ is singular. (iii) $A$ is diagonalizable if $B$ is so. (iv) $A$ and $B$ ...
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### In a matrix ring, no zero divisors may have an inverse

In a general ring with 1, a right (left) zero divisor cannot have a right (left) inverse. In a matrix ring over a field, a stronger condition is satisfied: a (right or left) zero divisor cannot have a ...
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### If $V$ is a vector space over a division ring $K$, and $A=\mathrm{End}_K(V)$, then every quotient ring of $A$ is a prime ring

Let $K$ be a division ring, let $V=V_{K}$ a vector space over $K$, and let $A=\mathrm{End}_{K}(V)$. Could anyone give me an idea of ​​how to prove that every quotient ring of $A$ is a prime ring?
I need to find $\varphi \in \operatorname{End}(\mathbb{R}[x])$ such that there's a function $\psi \in \operatorname{End}(\mathbb{R}[x]), \psi \neq 0$ such that $\psi \circ \varphi = 0$ but there's no ...