0
votes
1answer
35 views

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 $\mathbb{C}$. ...
0
votes
2answers
47 views

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. ...
7
votes
1answer
72 views

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 ...
1
vote
1answer
51 views

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 ...
1
vote
1answer
41 views

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 ...
2
votes
1answer
59 views

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 ...
0
votes
1answer
44 views

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 ...
3
votes
2answers
88 views

Showing that minimal polynomial has the same irreducible factors as characteristic polynomial

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 = ...
6
votes
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?
3
votes
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?
0
votes
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 ...
3
votes
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 ...
0
votes
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 ...
0
votes
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] = ...
7
votes
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 ...
2
votes
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 ...
1
vote
1answer
46 views

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 $ker(f)$ ...
2
votes
1answer
41 views

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 ...
1
vote
1answer
64 views

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 ...
2
votes
1answer
43 views

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 ...
0
votes
1answer
89 views

Ideal, not finitely generated

This exercise is driving me insane. I think there might be a mistake in it. Consider the ring $R$ of matrices of the form: $ R = \lbrace \begin{pmatrix} z & q_{1} \\ 0 & q_{2} \\ ...
0
votes
0answers
32 views

Are stably similar matrices similar?

Let $R$ be a ring and let $Gl_n(R)$ denote the set of invertible $n$ by $n$ matrices. Two matrices $A,B\in Gl_n(R)$ are called similar, if there exists another $P\in Gl_n(R)$ such that ...
0
votes
0answers
65 views

When is a vector space (over field $K$) also a ring (with subring $K$)?

(Apologies in advance for the very naive question. I'm just learning about all this. Also, for the sake of expedience, below I use the word "ring" when it would more correct for me to use ...
0
votes
0answers
42 views

Kernel of gcd and lcm of a number of polynomials when linear operator T is applied to them

I am trying to solve the following problem to later extend it to prove the primary decomposition theorem. I would like to show that if f1(x), f2(x), ..., fs(x) are polynomials in the ring F[x], with ...
5
votes
1answer
167 views

Generalization of Cayley-Hamilton

I'm having trouble following a proof of this generalization of the Cayley-Hamilton theorem: Suppose that $M$ is an $A$-module generated by $n$ elements, and that $\varphi \in ...
1
vote
0answers
37 views

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 ...
3
votes
1answer
76 views

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, ...
0
votes
1answer
39 views

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 ...
4
votes
3answers
163 views

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 ...
1
vote
2answers
117 views

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 ...
2
votes
1answer
37 views

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.
0
votes
1answer
66 views

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 ...
0
votes
1answer
45 views

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 ...
1
vote
1answer
50 views

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 $V/U$ ...
1
vote
0answers
52 views

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 ...
0
votes
1answer
81 views

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 $x$. ...
1
vote
1answer
63 views

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 ...
1
vote
1answer
64 views

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: $x^2=x$ $x^2-x=0$ ...
1
vote
1answer
39 views

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 ...
1
vote
1answer
43 views

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 ...
4
votes
2answers
113 views

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?.
3
votes
0answers
112 views

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 ...
6
votes
3answers
268 views

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 ...
5
votes
1answer
225 views

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 $B$ ...
7
votes
3answers
372 views

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 ...
2
votes
1answer
99 views

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?
1
vote
1answer
77 views

Finding ring endomorphisms.

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 ...
4
votes
3answers
255 views

Definition of principal ideal

This is a pretty basic question about principal ideals - on page 197 of Katznelson's A (Terse) Introduction to Linear Algebra, it says: Assume that $\mathcal{R}$ has an identity element. For $g\in ...
1
vote
2answers
74 views

Prove an inequality in a group ring

Let $$G=\bigoplus_{n\in\mathbb{Z}}\left(\mathbb{Z}/2\mathbb{Z}\right)_n$$ be a group, and for any $n\in \mathbb{Z}$, denote $\delta_n$ to be the element in $G$ with $n$-th coordinate $1$ and zero at ...
3
votes
1answer
112 views

Can we find subgroup of $(\mathbb{R},+)$ with order 2?

We used the following idea: first get a set of Hamel basis for $\mathbb{R}$, secondly, divide it into two parts such that one set of the Hamel basis forms a group, the other one is just the former one ...