For questions about rings which are not necessarily commutative and modules over such rings.

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5
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2answers
152 views

Where does the proof for commutative rings break down in the non-commutative ring when showing only two ideals implies the ring is a field?

We know in a commutative ring, if the only ideals are trivial and the whole ring, then the ring is a field, which is proved by every ideal is contained in a maximal ideal, which is proved by Zorn's ...
-1
votes
1answer
47 views

Does there exist a non-commutative ring of order $210$? [closed]

Does there exist a non-commutative ring of order $210$? One can construct rings of this order by taking products of $\mathbb Z/n\mathbb Z$, but those are commutative. Historical aside (not ...
3
votes
1answer
23 views

bilinear maps with respect to noncommutative rings

Consider a noncommutative ring with unity $R$, three left $R$-modules $M,N,P$ and a map $f\colon\;M\times N\to P$ such that: $ f(m+m',n)=f(m,n)+f(m',n)\\ f(m,n+n')=f(m,n)+f(m,n')\\ ...
1
vote
1answer
40 views

Gaussian for Grassmann variables

Let $(\theta,A\theta)=\theta_i A_{ij}\theta_j$ where $A$ is some $(2\times2)$ antisymmetric matrix. I want to generalize the following $$I(A) =\int d\theta_1d\theta_2~ ...
3
votes
1answer
40 views

Are finitely presentable modules closed under extensions?

If $0 \to A \to B \to C \to 0$ is an exact sequence of modules, and $A$ and $C$ are finitely presentable, then is $B$ finitely presentable? The answer is "yes" if we replace modules with groups, ...
0
votes
0answers
14 views

From progenerators to progenerators

I know that if $R$ is a ring (with identity) and $P$ is a progenarator right $R$-module (a f.g. projective generator) then $P/PJ$ is clearly f.g. when $J$ is the Jacobson radical of $R$; but, how ...
1
vote
1answer
21 views

trace of left/right multiplication

Let $A$ be a finite dimensional algebra over some field $k$. Then any element $a \in A$ defines two $k$-linear maps $a_l, a_r$ on $A$ by left and right multiplication, respectively. So there are two ...
1
vote
1answer
80 views

non-examples for Krull-Schmidt-Azumaya

I am looking for some rings $A$ and finitely generated $A$-modules $M$, where the conclusion of the Krull-Schmidt-Azumaya Theorem does not hold for $M$, i.e. where $M$ can be written as direct sums of ...
2
votes
2answers
39 views

A non-projective module

Let a ring with identity $R$ be decomposed as $S_1⊕\cdots⊕S_n$ (as a right $R$-module), where $S_i=e_iR$ with $e_i$ nonzero idempotents of $R$ adding up to $1$. If $J$ is the Jacobson radical of ...
3
votes
2answers
136 views

Why $J(M_n(R))=M_n(J(R))$ for any ring, where $J$ is the Jacobson radical of $R$?

I found this statement in an exercise: Show that $J(M_n(R))=M_n(J(R))$ for any ring $R$. This doesn't look like a trivial thing to me at all. The definition of Jacobson radical that my book ...
2
votes
2answers
44 views

A divisor of a unit is a unit?

Is it true that if $ab=u$ where $u\in U(R)$ is a unit of the noncommutative ring $R$, then $a,b\in U(R)$? If $R$ is commutative, then this can be seen by $$a(bu^{-1})=uu^{-1}=1=(bu^{-1})a,$$ but ...
1
vote
0answers
32 views

Inverse of a $2\times2$ matrix with noncommuting entries

I want to invert a $2 \times 2$ matrix with quaternionic entries. Since non-commutativity, the determinant is not well defined and I've seen in http://en.wikipedia.org/wiki/Quaternionic_matrix that, ...
0
votes
1answer
59 views

Nilpotent and invertible polynomials over noncommutative rings

Let $R$ be a noncommutative ring. 1) Prove or disprove: $a_0+a_1 x+\cdots+a_n x^n\in R[x]$ is nilpotent iff $a_i$ is nilpotent $\forall i$. 2) Prove or disprove: $a_0+a_1 x+\cdots+a_n x^n\in R[x]$ ...
0
votes
1answer
31 views

Global dimension of the center

Let $R$ be a ring. Must the global dimension of the centre $Z(R)$ of the ring $R$ always be atmost that of $R$ itself? I mean is it generally true that: $D(Z(R)) \leq D(R)$ (where D is the global ...
0
votes
0answers
22 views

Center of a ring projective?

If $R$ is a ring and $Z(R)$ denotes $R$'s centre, then when is $R$ projective as an $Z(R)$-module?
1
vote
1answer
30 views

Correct definition of bilinear(multilinear) maps over noncommutative rings

I wonder about the following: Let $R$ be a noncommutative ring with $\alpha, \beta \in R$ such that $\alpha \beta \neq \beta \alpha$. Let $M,N,P$ be left-$R$-modules. Why does it make problems to ...
0
votes
0answers
26 views

Ref. request for Linear algebra over noncommutative rings

This is not a real question but more a reference question. I am looking for introductory articles/blog entries/books that discuss the obstructions to do linear algebra over a non-commutative ring $R$. ...
0
votes
0answers
10 views

Showing that the operations defined on a right ring of fractions are well-defined

My problem comes from Goodearl & Warfield's "An Introduction to Noncommutative Noetherian Rings". Let X be a right Ore set of regular elements in a ring R. Define a relation $\sim$ on $R ...
3
votes
1answer
64 views

Why every central simple algebra has a splitting field

For central simple $F$-algebra $A$, where $F$ is a field, a field $E$ such that $F \subset E$ is called a splitting field if $A \otimes E$ is isomoprhic to $M_n (E)$ for some $n$. Why is algebraic ...
0
votes
0answers
21 views

Semisimple modules and short exact sequences

Though I suspect that this is a folklore result, I've been unable to find a proof by digital book searching. I also mention (my apologies) that I am new to noncommutative algebras theory and I am not ...
2
votes
1answer
25 views

Trace ideal of generator

Let $R$ be a unital ring. In Lam's "Lectures on modules on rings" (Theorem 18.8, p483), the following implication is stated for a right $R$-module $P$: $$\mathrm{tr}(P) = R \implies R \text{ is a ...
1
vote
1answer
30 views

Ideals in a polynomial ring over a skew field

I know that a polynomial ring over a field is a PID, does this property also hold for a polynomial ring over a skew field? Is there maybe something else that characterise the ideals in that ring ?
0
votes
0answers
23 views

Maximal and prime ideals of quaternions with integer coefficients

Let $R = \mathbb{Z} + \mathbb{Z}i + \mathbb{Z}j + \mathbb{Z}k$, the subring of $\mathbb{H}$ consisting of quaternions with integer coefficients. An exercise in Goodearl and Warfield's An Introduction ...
1
vote
0answers
39 views

Commuting exponentials of non-commuting matrices

For two non-commuting matrices $A,B \in M(2,\mathbb{K})$, $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, can be shown that: $$ e^C=e^{A+B}=e^Ae^B=e^Be^A \iff \begin{cases} ...
4
votes
1answer
73 views

Construction of noncommutative torus

In short, how do we get the formula for the NC torus? I find the equations in many places (including here) but I still have no idea for how this comes from the torus. If my understanding is correct, ...
6
votes
1answer
97 views

Schemes to the rescue?

I am reading chapter 1 of Gille and Szamuely's book Central Simple Algebras and Galois Cohomology, on quaternion algebras. In it they prove (remark 1.3.1 on p. 18) that the conic associated to a ...
1
vote
2answers
37 views

A concrete example of a unital noncommutative ring without maximal two-sided ideals

Whenever I say ideal in this question I'm talking about two sided-ideals. Does there exist a concrete example of a non-commutative ring with $1$ without maximal ideals? We know that if $R$ is a ...
5
votes
2answers
103 views

Defining algebras over noncommutative rings

A definition of "algebra" (that is, an associative algebra, in the sense of ring theory) generally requires a commutative base ring. But there are cases where it's reasonable to consider algebras ...
1
vote
1answer
31 views

$R/J(R)$ not semisimple Artinian

I search for a ring $R$ with Jacobson radical $J(R)$ such that $R/J(R)$ is not semisimple Artinian. Being a finitely generated module over itself, $R$ would have infinite hollow dimension due to ...
1
vote
0answers
25 views

Product notation $\prod$ when product does not commute [duplicate]

This is kind of a dubious question, but is the product notation $\prod$ often used in noncommutative rings? For example, if $M_i$ are matrices, I guess the common definition of $\prod$ is $$\prod_i ...
1
vote
0answers
37 views

Definition of Matrix Inverse [duplicate]

If I recall correctly, most of the references out there, including Wolfram Alpha, define the inverse of a square matrix $A$ to be a square matrix $B$ of the same dimension such that $AB=I$. But is ...
2
votes
1answer
537 views

Equivalent definitions of the Jacobson Radical

Noncommutative Algebra / Farb & Dennis defines the Jacobson Radical $J(R)$ of a ring $R$ as the intersection of all annihilators of simple $R$-modules (page 58). It is then claimed that $J(R)$ is ...
2
votes
1answer
88 views

Give an example of a maximal ideal in a noncommutative ring which is not prime

While trying to find an example, I came up with this: Since if $J$ is an ideal of a the ring $M_n(R)$, where $R$ is a commutative ring, then $J=M_n(I)$ for some ideal $I$ of $R$. IF I could show that ...
2
votes
1answer
58 views

Zerodivisors in polynomial rings over a non-commutative ring

Prove that if $f \in R[x]$ is a zero divisor then $\exists r(\neq 0) \in R$ s.t $rf=0$, where $R$ is a ring. I know that for $(a_0+a_1x+ \cdots ...
2
votes
1answer
31 views

Constructing noncommutative nilpotent rings of given index

When I read about algebra I often see a certain disregard for examples or perhaps a disregard for a reader whose knowledge of examples is limited. When I'm interested in a property $p$ of an algebraic ...
0
votes
0answers
24 views

A dual to essentially finitely generated notion

It is known that a (right) module $M$ over a ring $R$ has finite uniform dimension if and only if it is "essentially finitely generated", in the sense that there exists a finitely generated submodule ...
0
votes
1answer
16 views

Ideals of a skew polynomial ring where no positive power of the automorphism is inner

The exercise I'm trying to answer is as follows: Let $R$ be a ring, and $\alpha : R \rightarrow R$ an automorphism of $R$. Suppose that $R$ is simple and that no positive power of $\alpha$ is ...
0
votes
0answers
46 views

Identifying a Lie algebra from its universal enveloping algebra

Its been a while since I've worked on Lie algebras and I can't remember how to approach this problem: How do I identify the lie algebra (up to isomorphism) associated to a certain universal ...
1
vote
1answer
25 views

Projectiveness of direct product of projectives

It is well-known that a direct sum of modules is projective if and only if each summand is projective, and a direct product of modules is injective if and only if each of the modules are injective. ...
0
votes
0answers
27 views

A question about injection of a quotient in a direct summand

Let $M$ be a nonsingular right $R$-module in the sense that $Z(M)=\{m\in M : \operatorname{ann}(m)\text{ is essential in }R_R\}=0$. Could one find an $R$-monomorphism from the quotient $M/\bigcap N_i$ ...
0
votes
0answers
16 views

Goldie closure of the zero module

In "Lectures on Rings and Modules" by T.Y. Lam, it is claimed in Example (7.33) that the Goldie closure $cl(0)$ of the zero module $0$ on the ring $R=\left [\begin{array}\ \mathbb Z & \mathbb Z_2 ...
2
votes
1answer
50 views

$\mathfrak{u} \cdot R =R$ for every left ideal $\mathfrak{u}$ in a simple ring $R$

I am reading Noncommutative Rings by Lam. In his proof of the Wedderburn-Artin theorem (§1.3.11) he seems to use the following: If $\mathfrak{u}$ is a left ideal in a simple ring $R$, then ...
0
votes
1answer
31 views

Show that the probability that two randomly choosen elements from $R$ commute is at most $(5/8)$.

Let $R$ be a finite non-commutative ring. Show that the probability that two randomly choosen elements from $R$ commute is at most $(5/8)$. i know there is an answer in ...
3
votes
2answers
101 views

Do Lipschitz/Hurwitz quaternions satisfy the Ore condition?

The Lipschitz quaternions $L$ are the quaternions with integer coefficients and the Hurwitz quaternions $H$ are the quaternions with coefficients from $\Bbb Z\cup(\Bbb Z+1/2).$ A ring satisfies the ...
0
votes
1answer
49 views

A question in definition of group rings

In definition of a group ring $RG$ with elements $∑f_g g$ (where $g\in G$ and $f_g\in R$), are we supposed that $f_g$'s commute with $g$'s? I mean could we identify the above formal summation with $∑ ...
3
votes
1answer
42 views

Basis-free and noncommutative versions of the two-polynomials-over-ring problem (McCoy theorem etc.)

There is a rather canonical bunch of exercises in commutative algebra which tend to come up time and again on math.stackexchange: recently in #948010 and #83121, formerly in #227787 and #413788, and ...
14
votes
1answer
224 views

Rings with $a^5=a$ are commutative

Let $R$ be a ring such that $a^5=a$ for all $a \in R$. Then it follows that $R$ is commutative. This is part of a more general well-known theorem by Jacobson for arbitrary exponents ($a^n=a$), which ...
1
vote
0answers
37 views

Total ordering on the free group

The free groups can be totally (bi-)ordered. This paper shows how to do it (page 4). In short, you embed the group in multiplicative structure of the ring of power series in non-commuting variables, ...
1
vote
1answer
43 views

Jacobson radical of a matrix ring

I search for a way to prove that the Jacobson radical of $R=\left [\begin{array}\ \mathbb Z_4 & 2\mathbb Z_4 \\ 0 & \mathbb Z_4 \end{array} \right ]$ is $\left [\begin{array}\ 2\mathbb Z_4 ...
0
votes
0answers
35 views

When Leavitt path algebras are unital

I’ve been taking a seminar course on Leavitt path algebras, and our source material is rather laconic as far as explanations are concerned. I am attempting to justify (to myself) the following claim ...