1
vote
0answers
33 views

Complex matrices: looking for homomorphism

Let $\mathbb{C}$ denote the complex numbers, and let $M_2(\mathbb{R})$ be the ring of $2$ by $2$ matrices with real entries. Define a function $f:\mathbb{C} \to M_2(\mathbb{R})$ by $ f(a+bi) = ...
6
votes
1answer
409 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?
0
votes
1answer
36 views

Maximal Ideals of Matrix Ring

Let $D$ be a division algebra. I'm trying to show that all simple $M_n(D)$ modules are isomorphic to $D^n$. I know that $M_n(D)=D^n\oplus D_n \oplus \dots \oplus D^n$ (n terms). I have that if $S$ is ...
0
votes
0answers
18 views

Change of basis - free submodule

Let $R$ be a commutative ring and $(e_1, \dotsc, e_n)$ be a basis for $R^n$, the free $R$-module of rank n. Let $A$ be an $n \times n$ matrix with entries in $R$. Let $f_i = \sum \limits_{j=1}^n ...
3
votes
2answers
52 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
3answers
40 views

Show that $M_2(\mathbb{R})$ has no non-trivial two-sided ideals

In addition to the title question, I also want to find a non-trivial right ideal and a non-trivial left ideal of $M_2(\mathbb{R})$ . Attempt of title question: Suppose $\exists I\subset ...
7
votes
0answers
119 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 ...
1
vote
1answer
22 views

Is the Matrix ring over the a radical ideal a radical ideal of the matrix ring?

That is, if $I$ is an ideal filled with quasi-regular elements contained in ring $R$, are elements of the matrix ideal $M_n(I)$ also quasi-regular? It can be explicitly shown in the low dimensional ...
2
votes
2answers
93 views

What is the structure of matrix multiplication and minus?

Please note I have only little background im mathematics and I am working on formalizing theorems with theorem provers. This is very much a beginner question. Suppose I have matrices, where the ...
2
votes
1answer
103 views

Triangular Matrices and Simple Modules

Let $\Bbb{T}_n(k)=\{n \times n \text{ upper triangular matrices (which includes the diagonal entries)}\}$ I want to first express $\Bbb{T}_{n}(k)$ (as a $\Bbb{T}(k)$-module) as a direct sum ...
0
votes
1answer
56 views

How to prove that $J(M_n(R))=M_n(J(R))$?

How to prove that $J(M_n(R))=M_n(J(R))$? Here $M_n(R)$ is the ring of matrices of size $n^2$ over the ring $R$. And $J(M_n(R))$ is a two-sided ideal of the ring $M_n(R)$.
2
votes
1answer
90 views

for any ring $A$ the matrix ring $M_n(A)$ is simple if and only if $A$ is simple

Let integer $n\geq 1$. I have obtained that for any field $k$, the matrix ring $M_n(k)$ is simple, i.e., $M_n(k)$ contains no nonzero proper two sided ideals. Now I want to prove that: for any ring ...
1
vote
3answers
110 views

Cayley-Hamilton Theorem

I am trying to prove that all strictly upper triangular $n \times n$ matrices $A$, are nilpotent such that $A^n=0$. I am having trouble proving: $A$'s eigenvalues are all zero implies that ...
2
votes
1answer
79 views

Question concerning Eisenbud's theorem on matrix factorisations

I have the following question: Let $S$ be a commutative regular local ring and $\mathfrak{n}$ be its maximal ideal. Let $f\in\mathfrak{n}$ be a non zero-divisor in $S$ and let $m\geq 1$ ne a natural ...
0
votes
1answer
53 views

Free modules and free submodules. [duplicate]

Just finished my semester in advanced abstract algebra and there was one question that I could not answer. Let $R$ be commutative and let $(e_1,...,e_n)$ be a base for $R^{(n)}$. Put ...
4
votes
3answers
154 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
1answer
92 views

Conjecture about some group semiring representations ( and roots of unity ).

Let $\Bbb R_+=[0,\infty)$ be a semiring. $\Bbb R_+[C_n]$ is the group semiring formed by the semiring $\Bbb R_+$ and the cylic group $C_n$. Let $\Bbb R_+[X_n]$ be the polynomial semiring. ...
5
votes
2answers
177 views

$R$ has only one maximal ideal

Let $F$ be a field. Let $R$ be the set of all upper triangular matrices of the ring $M_{n}(F)$ with the property that its coefficients on the main diagonal are all the same. Prove that $R$ has only ...
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 ...
1
vote
1answer
78 views

Some basic questions about matrix rings and reversibility.

Neither commutative rings nor division rings are viable approaches to studying rings of matrices. However, there is a very cool notion of a reversible ring, which looks like it can fill this void. I ...
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?.
0
votes
2answers
103 views

Presentation of finite rings (fields)

One knows that every finite group is isomorphic to a subgroup of $\operatorname{GL}(n)$ for some $n$ large enough. Can every finite ring be represented by a ring of matrices, i.e., is every ring ...
1
vote
0answers
34 views

Can we always solve these linear algebra equations given rows of multiples?

We can find $n$ elements of multiplicative order $(n+1)$ modulo some large prime $p$, according to this question. Now I'm wondering if we can always perform linear algebra on the elements, as ...
3
votes
0answers
95 views

When powers of matrices are represented as a sum of integral matrices

There is given a ring $R$ and a subring $K$ with unit. We have a matrix $A$ of size $n$ over $R$. The characteristic of $R$ is $0$ or more than $n$. The statement is: If $A^m$ for any ...
3
votes
1answer
84 views

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 ...
5
votes
1answer
218 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 $B$ ...
1
vote
1answer
364 views

Show that every ideal of the matrix ring $M_n(R)$ is of the form $M_n(I)$ where $I$ is an ideal of $R$ [duplicate]

Suppose $R$ is a commutative ring. Show that every ideal of $M_n(R)$ is of the form $M_n(I)$ where $I$ is an ideal of $R$. I have spent 30 minutes on this question and I still got nowhere. Can ...
1
vote
1answer
46 views

Conjugacy class in matrix ring

Let $M_{2}(\mathbb{R})$ be the ring of $2\times 2$ matrices over the reals and $M_{2}(\mathbb{R})^*$ the set of invertible such matrices. Consider any $A \in M_{2}(\mathbb{R})$ such that $ A^{2}=-I$, ...
3
votes
1answer
65 views

Definition of $\text{GL}(n,R)$

How do one usually define the general linear group over a ring $R$, denoted by $\text{GL}(n,R)$. I was told in a paper that $\text{GL}(n,R)$ is a group, and I presumed that $$\text{GL}(n,R)=\{A\in ...
0
votes
1answer
83 views

Problem about a matrix on the ring $\mathbb Z$.

Suppose $A$ is a $m\times n$ ($n\geq m$) matrix on the ring $\mathbb Z$ of integers and the greatest common divisor of its $m\times m$ minor determinants is $1$. Prove that there is a $n\times m$ ...
4
votes
1answer
55 views

Finding an idempotent that satisfies certain conditions in a matrix ring.

I've been stuck on a problem, and I was wondering if anyone could help me out. The problem is: Let $R$ be the $2 \times 2$ matrix ring over the reals $\mathbb{R}$ of the form $$ \begin{bmatrix}a ...
5
votes
1answer
210 views

Polynomials in matrices with integer entries

I'm looking for references, if there is any, for this problem: Characterize all elements $a \in M_n(\mathbb{Z})$ for which we have $\mathbb{Q}[a] \cap M_n(\mathbb{Z})=\mathbb{Z}[a].$ Here, by ...
5
votes
3answers
186 views

How to construct a $2\times 2$ real matrix $A$ not equal to Identity such that $A^3=I$?

How to construct a $2\times 2$ real matrix A not equal to Identity such that $A^3$=I? There is a correspondence between the ring of complex numbers and the ring of $2\times2$ matrices (0 matrix is ...
2
votes
1answer
50 views

Is there an explicit way to determine $\mathrm{Mat}_n(R[X_1,\dots,X_m])\simeq\mathrm{Mat}_n(R)[X_1,\dots,X_m]$?

For a commutative ring $R$, let $\mathrm{Mat}_n(R[X_1,\dots,X_m])$ denotes the matrix ring with entries from $R[X_1,\dots,X_m]$, and let $\mathrm{Mat}_n(R)[X_1,\dots,X_m]$ denotes the polynomial ring ...
1
vote
1answer
63 views

Are there rngs whose rngs of matrices are commutative?

If $R$ is a unital ring and $M_{2\times 2}(R)$ is a commutative ring, then $R$ is a trivial ring because if $$\begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}=\begin{pmatrix}1 & 0 \\ 0 & ...
2
votes
0answers
125 views

Why is $M_{mn}(R)\simeq M_m(M_n(R))$? [duplicate]

Intuitively, it's not hard to believe that for a ring $R$, the matrix ring $M_{mn}(R)$ is isomorphic to $M_m(M_n(R))$. Taking a matrix in $M_{mn}(R)$ and turning the $n\times n$ blocks into single ...
4
votes
3answers
282 views

Rings with isomorphic proper subrings

Rings will be unital here but I don't require that subrings share the identity elements with superrings. I just accidentally came up with an example of a ring $R$ with a proper subring $S$ such that ...
5
votes
3answers
141 views

Is this action of $\mathbb F[[x]]$ on $\bigoplus_{i=0}^{\infty}\mathbb F$ natural?

The title of my question has a field $\mathbb F$ in it, but to make sure I'm not losing anything, I would like to introduce my question in full generality. But still, I will be happy with an answer in ...
0
votes
1answer
514 views

Smith Normal Form

Would the Smith Normal Form of the following matrix over $\mathbb Q[x]$ $$\begin{pmatrix}   (x+a)(x+b) & 0 & 0 &0 \\  0 & (x+c)(x+d) & 0 & 0 \\   0 ...
3
votes
1answer
272 views

Matrices over a ring

How might I find $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ where $a,b,c,d \in \mathbb Z[x]$ such that there does not exist $B, C \in M_2(\mathbb Z[x])$ such that $B^{-1}AC$ is ...
3
votes
1answer
243 views

Note on Ring Homomorphisms of Matrices Rings

Assume that $\mathbb{F}$ is a field, and let $\mathbb{M}_{t}\left( \mathbb{F}\right) $ be the ring of matrices of order $t$ over $\mathbb{F}$. Does there exist a non-trivial ring homomorphism from ...
18
votes
2answers
1k views

necessary and sufficient condition for trivial kernel of a matrix over a commutative ring

In answering Do these matrix rings have non-zero elements that are neither units nor zero divisors? I was surprised how hard it was to find anything on the Web about the generalization of the ...
9
votes
1answer
750 views

Do these matrix rings have non-zero elements that are neither units nor zero divisors?

Let $R$ be a commutative ring (with $1$) and $R^{n \times n}$ be the ring of $n \times n$ matrices with entries in $R$. In addition, suppose that $R$ is a ring in which every non-zero element is ...
5
votes
2answers
609 views

Is the determinant of a zero divisor zero?

Suppose that $A$ is a zero divisor in the ring of $(n\times n)$-matrices over the ring $R$. Is $\det(A) =0$ if $R$ is a field? Is $\det(A) =0$ if $R$ is an integral domain? It's not necessarily ...
3
votes
2answers
122 views

What is useful about commutativity/non-commutativity in practice

How is having commutativity or not having it useful for in practice? (whether it is for linear algebra, rings, basic airthmetic etc)
1
vote
1answer
202 views

How to find matrices with given commutator

Consider $M_2(\mathbb{Z})$. Is it possible to find two matrices A,B such that their commutator AB - BA equals a given matrix C? Is there any chance to characterize all possible occuring commutators in ...
8
votes
4answers
529 views

“weird” ring with 4 elements - how does it arise?

For no real reason, I did a classification of (associative, with a 1) rings with less than 8 elements (they all happen to be commutative). Most of the rings I got were of a type I knew - namely: ...
5
votes
2answers
160 views

Rings of matrices

Let $ A\in {\mathbb{F} }^{n\times n} $ be a fixed matrix. The set of all matrices that commute with A forms a subring of ${\mathbb{F} }^{n\times n}$. Is any subring of ${\mathbb{F} }^{n\times n }$ ...