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

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21 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?
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1answer
29 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 ...
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21 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 $R$. ...
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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 ...
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1answer
22 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 ...
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1answer
59 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 ...
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0answers
20 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 ...
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1answer
23 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 ...
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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 ...
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0answers
37 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} ...
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1answer
29 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 ?
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2answers
34 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 ...
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1answer
28 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 ...
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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 ...
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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
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1answer
69 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 ...
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1answer
56 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 ...
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1answer
29 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 ...
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0answers
23 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 ...
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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. ...
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0answers
45 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 ...
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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 $M/\bigcap N_i$ ...
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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 ...
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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 ...
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1answer
40 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 ...
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1answer
47 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 $∑ ...
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1answer
15 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 ...
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0answers
36 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, ...
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1answer
38 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 ...
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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 ...
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1answer
217 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 ...
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0answers
24 views

Every right principal ideal non-emptily intersects the center — what is that?

This is a follow-up to Do Lipschitz/Hurwitz quaternions satisfy the Ore condition? Jyrki Lahtonen answered the question in the positive by noticing that every right principal ideal in either ring has ...
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2answers
99 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 ...
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2answers
86 views

Is Planck's constant a mathematical or a physical constant?

Planck's constant $h$ came out of his considerations of black body radiation and features prominently in quantum physics. Recently I came across the statement that $-i\hbar = pq - qp$ for elements ...
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0answers
33 views

Endomorphism rings of a $k$-algebra

Let $k$ be a field and $R$ be the $3$-dimensional $k$-algebra $\left [\begin{array}\ k & k \\ 0 & k \end{array} \right ]$. I have two conjectures: (1) Is any simple right $R$-module has ...
2
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0answers
124 views

Learning roadmap in Algebra

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas: a) Commutative Algebra b) Field Theory and Galois Theory c) Homological Algebra My question is ...
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48 views

An error in the book “noncommutative ring” writed by Herstein

I'm reading the book "noncommutative ring" writed by Herstein. In the page 15, the author says that Let $F$ be a field and $A$ is an algebra over $F$. Let $\rho$ be a maximal regular right ideal ...
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1answer
26 views

A uniform module as an intersection

Let $R$ be a semiprime, nonsingular ring with finite Goldie dimension u.dim $R_R$. (Nonsingularity means here that $Z(R_R)=0$, where $Z(R_R)$ is the set of elements $x$ of $R$ with $ann(x)$ is ...
4
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1answer
72 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, ...
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0answers
25 views

Considerations for moving a function inside or outside of an integral

Excluding the possibility that $A(t)$ is the limit of a sequence, are there any special considerations I should be concerned with regarding the following assertion: Let $A(t)$ be an $n\times n$ ...
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1answer
30 views

Extension of an $R$-homomorphism as a sum

I want to solve this problem: Let $M$ be an injective module over a ring $R$ with identity, and $f$, $g$, $h$ are elements of the ring of $R$-endomorphisms $\operatorname{End}(M)$ such that $\ker ...
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1answer
79 views

A central division algebra is not its commutator

In looking at old qualifying exam questions, I've come upon a question that has me stumped. Let $A$ be a central division algebra (of finite dimension) over a field $k$. Let $[A,A]$ be the ...
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2answers
26 views

A direct limit concerning some homomorphisms

In an algebra text there is the following argument I am stuck in the last part of which: "Let $f:B→C$ be an epimorphism in the category of $R$-modules, and $D=∑_{n=1}^∞c_nR$ be a countably generated ...
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1answer
49 views

Isomorphism between $R$ and its dual space

Let $R$ be a finite dimensional algebra over a field $K$. If $f$ is an $R$-module monomorphism from $R$ to the dual $K$-space $\operatorname{Hom}_K(R,K)$ why it is onto? Thanks!
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1answer
43 views

Existence of a maximal submodule

Let $R$ be a left Artinian ring and let $M$ be a nonzero left $R$-module. Prove that $M$ has at least one maximal submodule. I don't really know where to start. Any hint? Thanks!
2
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1answer
51 views

Von Neumann regular but not self-injective ring

I want an example of a von Neumann regular ring which is not self-injective. My thanks go to anybody answering.
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1answer
43 views

Another description of injective hull

Let $I$ be an injective module containing a module $M$, let $M_1$ be a submodule of $I$ maximal with respect to the property that $M_1∩M=0$, and let $M_2$ be a submodule of $I$ containing $M$ maximal ...
2
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1answer
64 views

Showing the Weyl algebra is simple.

Let $R$ be a ring with $1$, which contains $\mathbb{Q}$, and generated over $\mathbb{Q}$ by two elements $x,y$ such that $yx-xy=1$. Show that $R$ is simple. What i did? Certainly $x, y \in R$ as ...
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0answers
27 views

Example of a ring of finite global dimension with flat qu0tient

I've been thinking about this for quite a while but I cannot seem to find an example of If $k$ is a commutative ring of finite global dimension and I'm looking for a strictly not-commutative ...
2
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3answers
65 views

Contrasting definitions of bimodules? An illusion?

Recently my definition of a bimodule over a $k$-algebra has been challenged and I believe both definitions to be equivalent, am I wrong? Notation: $k$ is a commutative ring and $A$ is a (unital ...