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

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Is $R_\mathfrak{p}$ local when $R$ is noncommutative?

This may seem like a basic question but I can't seem to find any good texts on the matter (perhaps someone can suggest any?). Basically I have a noncommutative ring $R$ with prime ideal $\mathfrak{p}$....
4
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2answers
59 views

What algebra is generated by $\mathrm{O}(2)$?

The unit complex numbers can be identified with the $2 \times 2$ special orthogonal matrices $\mathrm{SO}(2)$. The problem with $\mathrm{SO}(2)$, however, is that its not closed under $\mathbb{R}$-...
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0answers
9 views

Indecomposable Modules over an algebra

If I have $K$ subfield of $F$ and $A$ an $K-$algebra (associative ,finite dimensional and with with unity). And take an ciclic indecomposable $A-$module $M=A\lambda$. Then $A\otimes_K F(\lambda\...
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3answers
67 views

Baker-Campbell-Hausdorff/Zassenhaus formula to first order in one matrix

Is there a closed-form expression for the term of $e^{t(c \hat{X} + d \hat{Y})}$ that is first-order in $d$, where $t$, $c$, and $d$ are scalars and $\hat{X}$ and $\hat{Y}$ are finite-dimensional ...
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0answers
13 views

Making a multivariate polynomial monic in one of its variables

I apologise in advance for the general nature of this question. Suppose we have a non-commutative ring $R$ that is relatively well-behaved as non-commutative rings go (I was thinking of $R$ being the ...
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0answers
49 views

Simplify Series composed by Noncommutative Matrices

Problem I need to find a simpler formula for the following series: S = $\sum_{a=1}^{\infty} \frac{1}{a} \sum_{b=1}^{a} X^{b-1}MX^{a-b} = \sum_{a=1}^{\infty} \frac{1}{a} X^{a-1} \sum_{b=0}^{a-1} X^...
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2answers
29 views

Extension of a finite field to a finite non commutative ring

Can a finite field be extended to non-commutative finite rings so that not all elements of the field commutes with the elements of the ring? I have been trying this taking the examples of matrices.
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24 views

Trivial extension of an opposite algebra

Suppose that $A$ is a finite dimensional $K$-algebra, where $K$ denote an algebraically closed field. Call $DA=Hom_k(A,k)$. $DA$ admits an $A$-$A$-bimodule structure in the obvious way. The trivial ...
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1answer
26 views

Is the tensor algebra functor a strong monoidal functor?

Let $K$ be a commutative ring. Is it true that $T(V \oplus W) \cong T(V) \otimes_K T(W)$ as $K$-algebras for any $K$-modules $V$ and $W$? The reason I ask is that I have heard it mentioned (I don't ...
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0answers
20 views

When does the unit of universal enveloping algebra adjunction fail to be injective?

The Poincaré–Birkhoff-Witt (PBW) theorem implies that if $K$ is a commutative ring and the Lie $K$-algebra $\mathfrak{g}$ is a free $K$-module, then the unit $\eta_{\mathfrak{g}}: \mathfrak{g} \to U (\...
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1answer
43 views

Adjointness of Hom and Tensor for non commutative ring

Let $M$ be an $(A,B)$-bimodule, let $N$ be a $(B,C)$-bimodule, and let $K$ be an $(A,C)$-bimodule. Then $$\operatorname{Hom}_C(M \otimes_{B}N,K) \cong \operatorname{Hom}_B(M,\operatorname{Hom}_C(N,K))$...
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1answer
35 views

If the ring $M_n(D)$ is a $k$-algebra, is $D$ a $k$-algebra?

Let $k$ be a field, let $D$ be a division ring. Assume the matrix ring $A = M_n(D)$ is endowed with some $k$-algebra structure compatible with its ring structure (namely, for all $\lambda \in k$ and ...
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0answers
38 views

Idempotents central or not?

Let $R$ be a nil-clean ring with unity such that $R/J(R)$ is reduced, where $J(R)$ is the Jacobson radical of $R$. Is it true that $R$ is abelian, i.e. the idempotents are central? (By nil-clean I ...
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0answers
70 views

Name these symmetric polynomials over non-commutative variables…

Given a set of $N$ non-commutative variables $x_k$. Is there a special name for symmetric polynomials of homogenous degree $d$ of the form that all $x_k$s appear with exponent at most $1$ at a time? ...
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2answers
99 views

Nilpotent or non-Nilpotent Jacobson Radical

Let $R$ be a ring with identity element such that every ideal of which is idempotent or nilpotent. Is it true that the Jacobson radical $J(R)$ of $R$ is nilpotent? If $R$ is Noetherian and $J(R)$ is ...
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1answer
45 views

An Equality in Ring Theory

Let $R$ be a ring with identity (commutative or not), and assume $J(R)$ to be its Jacobson radical. Let $e\in R $ be an idempotent and $x\in J(R)$ be a given fixed element. I am searching for ...
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0answers
18 views

What are the fixed points of $\alpha^n-\mu_j$ for a fixed $j$?

Let us consider the polynomial ring $\Bbb C[x_1,...,x_s]$ and $\alpha(x_i)= x_i + \mu_i$ where $\mu_i \in \Bbb C$ are not all zero. Then $\alpha \in \mathrm{Aut}(\Bbb C[x_1,...,x_s])$. What are ...
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0answers
44 views

A problem with infinitely many eigenvalues on a finite dimensional vector space

I want to develop some theory before posing the problem. Kindly stay with me. Consider $ Aut (k[x_1,...,x_n])$ where $k$ is an algebraically closed field, you can take $k=\Bbb C$. $\alpha \in Aut(k[...
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1answer
36 views

If $a$ and $b$ are elements in a ring with $a^n=b^n$ and $a^m=b^m$ then $a=b$

I was doing the first exercises from the book Exercises in Basic Ring Theory by G. Călugărescu and P. Hamburg and I found one whose solution isn't quite clear to me. Ex. 1.4 If $a$, $b$ are ...
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1answer
21 views

Empty singular submodule

I search for a module $M$ with its singular submodule $Z(M)$ the empty set, i.e. for every element $m$ of $M$ the annihilator of $m$ in $R$ is not essential, say, as right $R$-module; or proving that ...
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0answers
32 views

Is there a nonzero polynomial in $\mathbb{H}[x]$ which vanishes in all $\mathbb{H}$?

I know that over any infinite field $F$ there are no nonzero polynomials in $F[x_1,\cdots,x_n]$ which vanish in all $F^n$ (Proof is by induction with basis step given by the fact that polynomials in $...
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0answers
32 views

Finding an essential submodule

Let $R$ be a commutative ring with unity, and let $M$ be a unitary faithful $R$-module. Assume the annihilator $A$ of $r\in R$ is essential in $R$ (as an $R$-module). I search for an essential ...
2
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1answer
37 views

Socles and factors

Let $A$ be a finite dimensional algebra over a field $K$ and let $M$, $M'$ and $N$ be $A$-modules. Suppose that $M'\subseteq M$ and assume that $N$ has simple socle. Let $f: M \longrightarrow N$ be ...
2
votes
1answer
28 views

Singular ideal containing a given nilpotent ideal

Let $R$ be a ring with identity, and $Z(R_R)$ be the singular ideal. Is it true that any nilpotent ideal of $R$ lies in $Z(R_R)$? It is well known that any central nilpotent element would belong to ...
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0answers
25 views

Inverse elements in a certain monoid

Let $R$ be a ring with unity, and $Z(R_R)$ be its right singular ideal, i.e. the set of elements of $R$ whose right annihilators are essential in the right module $R_R$. My question: If $x\in Z(...
4
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1answer
57 views

If $D$ is a triangulated category, and $E_i$ is a set of generators, then $D$ is equivalent to $D(End(\oplus E_i))$?

I am looking for a result along the lines of the following statement: If $D$ is a triangulated category, and $E_i$ is a set of generators (every object can be obtained up to isomorphism by shifts and ...
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1answer
47 views

Part of a proof that a left $R$-module $M$ is cyclic and every nonzero element generates $M$ if and only if $M\cong R/I$ for a maximal left ideal $I$.

I'm trying to follow this proof from Noncommutative Algebra by Farb. The theorem is that the following are equivalent for a left $R$-module $M$: $(1)$ $M$ is simple $(2)$ $M$ is cyclic and generated ...
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0answers
10 views

A modularity condition

Let $R\subseteq S$ be rings with unity and $X$ ,$Y$ be subsets of $S$ with $X$ an ideal. If $S=X+Y$ what conditions should be held to infer the equality $R=(X∩R)+(Y∩R)$? I think that if we ...
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1answer
27 views

Singular ideals and rings

In Lam's book, Corollary (7.4)(2) says that for a nonzero ring $R$ we have $Z(R_R)≠ R$, where $Z(R_R) $ stands for the singular ideal of $R$.. But, some nonzero commutative rings are "singular" in the ...
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2answers
91 views

Best texts on supermathematics for a mathematician?

I'm an undergraduate who's doing some summer mathematics research, and it looks like I need some information on Berezenians and supermatrices as well as supermathematics in general. The only text I ...
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1answer
19 views

Cyclic algebras of degree $4$ and period $2$

Recall that if a field $k$ has a primitive $n$-root of unity $\omega$, then the cyclic $k$-alegbras of degree $n$ (ie of dimension $n^2$) have the following familiar presentation : they are generated ...
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0answers
47 views

How to understand the formula $\Delta(x_1 \otimes x_2) = \Delta(x_1)\Delta(x_2)$?

In the webpage, it is said that the comultiplication on the tensor algebra $TV$ is defined as follows. \begin{align} \Delta(x_1\otimes\dots\otimes x_m) = \sum_{p=0}^m \sum_{\sigma\in\mathrm{Sh}_{p,m-p}...
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1answer
17 views

Relationship between idempotents in semisimple ring.

Let R be a semisimple ring with identity. If $e$ and $e'$ are idempotents in R such that $Re \simeq Re'$ then there exists $a \in R^\times$ such that $e' = aea^{-1}$ Attempt I know from a previous ...
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1answer
17 views

What is the reduced norm map?

This is a basic question about the reduced norm homomorphism. Let $A$ be a central simple $K$-algebra and $P$ a f.g. projective $A$-module. I know that $\operatorname{End}_A(P)$ is also a central ...
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1answer
76 views

How to study for ring theory?

I want to study the theory of rings because it is used when I study representation theory. Here, a ring is not necessarily commutative and doesn't necessarily has unity. I know that there are a few ...
4
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1answer
46 views

Central idempotents of a ring which has only one simple module up to isomorphism

Let $R$ be a ring such that any two simple $R-modules$ are isomorphic. Show that $R$ has no nontrivial central idempotent. I know how to prove this for a simple ring $R$, since a central idempotent ...
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2answers
60 views

Invariant dimension property of a ring $R$ which admits a homomorphism to a division ring $D$

Let $R$ be a ring which admits a homomorphism to a division ring $D$. I know that if the homomorphism is surjective, then the ring has invariant dimension property. But if the homomorphism is not ...
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3answers
121 views

Identity of a ring as two different sums of idempotents

Let $R$ be any ring with identity $1_R$. Prove that if there exist idempotents $e_1,..., e_n, e'_1,...,e'_n \in R$ such that $$ 1_R = e_1 +...+ e_n = e'_1+... e'_n$$ then the following conditions ...
2
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1answer
20 views

Upper Nilradical of a Ring

If we define the upper nilradical of a ring as the sum of all nil ideals of the ring, how could we deduce from just this definition that this is a nil ideal? Thanks!
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1answer
59 views

Rings in which $ab=0$ implies $axb=0$

I'm sure there must be some standard term for (not necessarily commutative) rings $R$ in which $ab=0$ implies $(\forall x)\, axb=0$ (for example, this is the case if $R$ is commutative or is a domain)....
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1answer
28 views

Show that $\operatorname{End}_A(P)$ is a central, simple $K$-algebra if $P$ is a f.g. projective $A$-module

Let $A$ be a central, simple $K$-algebra and let $P$ be a finitely generated projective $A$-module. I want to show that the endomorphism ring $\operatorname{End}_A(P)$ is also a central, simple $...
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1answer
70 views

A question about tensor product and multiplication?

Let $k$ be a commutative ring, $A$ be an algebra over $k$. The tensor product $A\otimes A$ is over $k$. If $\sum_{i} a_i \otimes b_i=\sum_j c_j\otimes d_j$, where $a_i,b_i,c_j,d_j\in A$, I wonder if $\...
2
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1answer
27 views

Direct Summands of PI Rings as Right Ideals

Is any direct summand of a PI-ring (polynomial identity ring) necessarily idempotent as a right ideal? The answer is yes for a special case of PI-rings, namely any direct summand of a commutative ...
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0answers
31 views

Any characterization of rings over which submodules of left free modules are free?

Let $R$ be a ring (with identity, but not necessarily commutative). If given the presumption that for any left free $R$-module $M$, every submodule of $M$ is free, what can we say about $R$? ...
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1answer
24 views

Sum of nil right ideals as an ideal

I have two questions: 1) If $S$ is the sum of all right nil ideals of a ring $R$ (with unity), is it true that $S$ is a two-sided ideal? It is clear for me that $S$ is a right ideal (and it is nil if ...
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1answer
21 views

$N(R)$ when $R$ is a P.I. ring

The set of nilpotent elements $N(R)$ of a ring $R$ with identity is not necessarily a right ideal (or even a subgroup) as it is seen in the ring of $2×2$ matrices over $\mathbb Z$. But, my question ...
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1answer
56 views

$R$ is isomorphic to a direct product of matrix rings over division rings

Suppose as rings, $R$ is isomorphic to a direct product of matrix rings over division rings, that is $R=R_1 \times ... \times R_n$ where $R_i$ is a two-sided ideal of $R$ and $R_i$ is isomorphic to ...
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1answer
27 views

Is the field of fractions of $F[x_1, \dots, x_n]$ a Noetherian Weyl algebra module

Let $F$ be a field of characteristic zero. Let $D_n$ be the Weyl algebra, i.e., $D_n \subset \mathrm{End}_F(F[x_1, \dots, x_n])$ is the submodule generated by $x_i$ and $\partial_i$, $i = 1, \dots, n$...
1
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1answer
76 views

Show that $I$ is an ideal

Let $R$ be a ring and $I\subseteq R$ the only maximal right ideal of $R$. I want to show that $I$ is an ideal. To show that $I$ is an ideal, we have to show that $I$ is a left ideal, right? How ...
0
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0answers
47 views

quantum matrices, quantum determinant

Homework, except I'm completely clueless, so if someone could potentially point me to similar worked examples or help explain this one step at a time it would be much appreciated. Could you explain ...