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

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0answers
36 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[...
6
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2answers
119 views

Books on Rings without Identity

I was just wondering if anybody knows of any good books or articles that study rings (and algebras) without (or not necessarily with) identity. I have gone through Thomas Hungerford's Algebra textbook ...
2
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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 ...
1
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1answer
18 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
31 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
28 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
votes
1answer
26 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 ...
2
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1answer
34 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 ...
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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(...
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0answers
54 views

free associative algebras as tensor product

Let $A,B$ be $k$-algebras with $k$ a commutative ring. Suppose that the free product $A \cdot_k B$ is a free associative $k$-algebra of rank $n$. Does it imply that $A,B$ are both free associative $k$-...
2
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2answers
88 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 ...
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 ...
1
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1answer
43 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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1answer
134 views

An example of Von Neumann regular ring with non-zero nilpotent elements. [closed]

Give an example of a Von Neumann regular ring having non-zero nilpotent elements.
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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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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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0answers
46 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 ...
1
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1answer
18 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 ...
3
votes
1answer
480 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 ...
1
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1answer
16 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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0answers
45 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}...
6
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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 ...
0
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1answer
18 views

Zassenhaus formula exponents

found Zassenhaus formula for noncommutative $X,Y$ $$e^{t(X+Y)}= e^{tX}~ e^{tY} ~e^{-\frac{t^2}{2} [X,Y]} ~ e^{\frac{t^3}{6}(2[Y,[X,Y]]+ [X,[X,Y]] )} ~ e^{\frac{-t^4}{24}([[[X,Y],X],X] + 3[[[X,Y],X],Y]...
2
votes
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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1answer
66 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 $\...
1
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1answer
16 views

Filtered Colimit of associative $k$-algebras that are domains

Let $C$ be a filtered subcategory of the category of commutative algebras over a fixed field $k$ whose objects are all integral domains. Then the colimit of the obvious diagram is an integral domain. ...
3
votes
1answer
66 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 ...
0
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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 ...
4
votes
1answer
41 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 ...
2
votes
1answer
19 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!
1
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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 ...
0
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1answer
62 views

Lie rings: reference request

Dear friends: I am looking for a modern reference for Lie rings (In particular, I would like to have nice references for the structure of Lie ideals), let it be lecture notes or a book, in the sense ...
0
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0answers
58 views

Simple algebra that is not a simple ring

Maybe this question is trivial, however I'm not acquainted with non-commutative stuff. Here it's written that a simple algebra may not be a simple ring. The definition of simplicity are the usual ...
4
votes
1answer
57 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)....
2
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1answer
26 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 ...
0
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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 $...
5
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1answer
65 views

Why is $Ind^G_H(M)=Ind^{G/H}_{\{e\}}$?

I was looking at some representation theory notes and found the following statement: $Ind^G_H(V)=\mathbb{C}[G]\otimes_{\mathbb{C}[H]}V=\mathbb{C}[G/H]\otimes_\mathbb{C} V$. Now, this makes intuitive ...
2
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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
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 ...
3
votes
1answer
117 views

Compute the Jacobson radical of the group ring $\mathbb{F}_2S_3$.

Compute the Jacobson radical and the maximal semisimple quotient of the group ring $\mathbb{F}_2S_3$ of the symmetric group on three letters over the field with two elements, and compute the same ...
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2answers
2k views

Finite rings without zero divisors are division rings.

How can I prove this: Finite rings without zero divisors are division rings. I know how to prove it when the ring has $1$, but I have no idea if my ring needs to have an unity.
4
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2answers
77 views

Is the tensor product of non-commutative algebras a colimit?

For $R$ a commutative ring, the tensor product of $R$-algebras is the coproduct in the category of commutative $R$-algebras. In the noncommutative case it is no longer the coproduct in the category of ...
1
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1answer
55 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 ...
0
votes
1answer
26 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 ...
1
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1answer
30 views

Sum of nil right ideals

Is an arbitrary sum of nil (nilpotent) right ideals a nil (nilpotent) right ideal? If $I=\sum I_i$ is a sum of nil ideals then each element $x$ of $I$ is a finite sum $x=x_1+...+x_n$ of elements $x_k\...
13
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1answer
321 views

Is this a characterization of commutative $C^{*}$-algebras

Assume that $A$ is a $C^{*}$-algebra such that $\forall a,b \in A, ab=0 \iff ba=0$. Is $A$ necessarily a commutative algebra? In particular does "$\forall a,b \in A, ab=0 \iff ba=0$" ...
3
votes
1answer
59 views

Algebraic delta functions

Let $D_n$ be the Weyl algebra: $D_n \subset \mathrm{End}_\mathbf{C}(\mathbf{C}[x_1, \dots, x_n])$ is generated by $x_i$ and $\partial_i$, $i = 1, \dots, n$. My professor says the $D_n$-module $$ M = \...
4
votes
1answer
42 views

Descending chain condition and an identity

Let $R$ be a ring with descending chain condition on right ideals. Suppose $l(R)=0$ (the left annihilator of $R$) and $\exists c\in R$ with $r(c)=0$ (the right annihilator of $c$). Show that $R$ has ...
0
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0answers
30 views

A question about ring has 1 stable range

I got this problem from my professor.It states that if $D$ is a division ring then ring of matrices $M_{n}(D)$ has 1 stable range. A ring called has $1$ stable range if we get $Ra+Rb=1$ for some $a,b ...