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

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+50

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 ...
4
votes
2answers
96 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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0answers
26 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
votes
1answer
31 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 ...
1
vote
2answers
45 views

Invariant dimension property of a ring $R$ 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 surjective, is it still true that $R^n$ equivalent to $R^m$ as $R-$modules ...
2
votes
1answer
17 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
23 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
61 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 ...
3
votes
1answer
49 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 ...
2
votes
1answer
23 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 ...
0
votes
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
votes
1answer
61 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
votes
0answers
29 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 $R$? ...
1
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1answer
19 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
87 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 ...
8
votes
2answers
1k 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
votes
2answers
73 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
vote
1answer
44 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 ...
3
votes
1answer
475 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 ...
0
votes
1answer
24 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, ...
1
vote
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? ...
1
vote
1answer
29 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 ...
0
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0answers
37 views

quantum matrices, quantum determinante

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 ...
0
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0answers
10 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] + ...
12
votes
1answer
301 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
40 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
28 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 ...
2
votes
1answer
27 views

Defining a radical of a module axiomatically

In these notes by Richard Vale, he approaches the notion of a radical of a module axiomatically, by saying the following. A radical is an assignment, to each R-module $M$, of a submodule $\tau(M) ...
2
votes
2answers
55 views

Is there a ring which satisfies $xy=1$ and $yx\neq 1$ [duplicate]

I checked a lot of examples of non-commutative rings that came to my mind, but they weren't helpful. In particularly it's not the case for ring of matrices because of the multiplicity of the ...
4
votes
1answer
30 views

If $R$ is a domain and $M_n(R)$ is semisimple, then $R$ is a division ring.

From Lam's A First Course in Noncommutative Rings, section 1.3. Let $R$ be a domain (EA: that is, a ring without zero divisors) such that $M_n(R)$ is semisimple. Show that $R$ is a division ring. ...
1
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1answer
51 views

Axioms of noncommutative rings [duplicate]

Is there an abelian group $R$ with multiplication operator with this properties ? (i) $a(bc)=(ab)c$ (ii) $a(b+c)=ab+ac$ , $(b+c)a=ba+ca$ And a unique element $e$ s.t (iii) $ea=a\quad ...
1
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0answers
23 views

Closed form of Baker Campbell Hausdorff theorem with cyclic bracket structure

I would like to know if there exists a closed form of the Baker Campbell Hausdorff theorem subject to the conditions that $[x,[x,y]] \sim x$ and $[y,[x,y]] \sim y$. The simple cases that I know a ...
0
votes
0answers
25 views

Classical ring of quotients.

I am studying classical rıng of quotients and ore localization.I can't understand how to use universal property and how to find ring of quotients any given ring R.For example;Let be $R= ...
5
votes
1answer
366 views

Localization of a non-commutative ring

Let $A$ be the non-commutative ring given by $$ A=\mathbb{C}\langle x,y,z \rangle /(xy=ayx,yz=bzy,zx=cxz) $$ for some $a,b,c\in \mathbb{C}$. What is the localization $A_{(x)}$ of A with respect to the ...
1
vote
1answer
113 views

Prove if a non-trivial ring $R$ has a unique maximal left ideal $J$ , then $J$ is two-sided and is also the unique maximal right ideal in $R$.

If a non-trivial ring $R$ has a unique maximal left ideal $J$ , then $J$ is two-sided and is also the unique maximal right ideal in $R$. I can prove that it is two sided, but I can't prove that it is ...
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0answers
38 views

Example of noncommutative ring in which all nonzero elements are invertible [duplicate]

Example of noncommutative ring in which all nonzero elements are invertible. I was thinking along Matrices in $(2,\mathbb Z$), but some nonzero matrices are not invertible. Can someone provide ...
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0answers
38 views

Solvability of an Equation in a Noncommutative $\mathbb Q$-algebra

Let $A$ be a noncommutative $\mathbb Q$-algebra, and let $a\in A$. If we know that $a$ has a left inverse in $A\otimes \mathbb Q_\ell$ for a prime $\ell$, then we can conclude that $a$ is already ...
-1
votes
1answer
21 views

Multilinear map with algebra

I write this question "Introduction to Noncommutative Algebra" written by Bresar."6.1 Free Algebra p.139" I didn't show that how f is multilinear.Every F-algebra A,the map ...
0
votes
1answer
22 views

I cant understand free algebra or free ring.

I want to write as a set any free algebra or ring.For instance, $K[t]/(t^{2})=${$ g(t)+(t^{2}):g(t) \in K[t]$}$\hspace{1.9cm}=${$at+b+(t^{2}):a,b \in K$} $(t^{2})$={$f(t)t^{2}:f(t)\in K[t]$} As above ...
4
votes
1answer
63 views

Why are these definitions of groups of central type equivalent?

Let $G$ be a finite group. In the celebrated paper of Howlett and Isaacs, On Groups of Central Type, Math. Z. (1983)., the group $G$ is called to be of central type if $G$ has an irreducible ...
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1answer
75 views

Polynomials in two noncommuting variables

Let $F$ be a field. Let's suppose we have this ring $$F\langle x,y\rangle$$ when $xy\neq yx$ and the elements take this form $$p\langle x,y\rangle=\sum_{i=1}^n c_i$$ where $c_i=x^{a_{1i}} y^{b_{1i}} ...
4
votes
1answer
58 views

GCD of matrices [closed]

Given $A=M_1SN_1$ and $B=M_2SN_2$ where all $M_1,M_2,S,N_1,N_2\in\Bbb Z_{\geq0}^{n\times n}$ are all symmetric full rank is there a procedure to extract $S$ from $A$ and $B$ using gcd like operations? ...
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0answers
16 views

Indecomposable module and injective hull

If an $A-$module $M$ over finite dimensional algebra is indecomposable, then it has an injective hull indecomposable. Why?? Thanks!
0
votes
1answer
42 views

Reduced Trace, Central Simple Algebra

Let $A$ be a central simple $K-$algebra and $A = M_n(D^{o}), Z(D) = K, [D:K] = m^2$. Let $\psi(a)$ be the reduced trace of an element of $A$. I am not sure if the definition I ...
0
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0answers
12 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. ...
1
vote
1answer
21 views

Rings of finite uniform dimension

Let a ring $R$ has finite Goldie dimension, say, as left $R$-module. Could it be deduced that $R$ is $I$-finite in the sense that $R$ does not contain an infinite subset of non-zero orthogonal ...
0
votes
0answers
18 views

Assumptions of Skolem-Noether Theorem

Let $k$ be a field, $A,B$ simple $k$-Algebras and such that $Z(B)=k$. My lecture notes say that if $A$ is finite over $k$, then all $k$-homomorphisms $A\longrightarrow B$ are conjugate. Is this ...
0
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0answers
12 views

Representation of left ideals of the matrix ring [duplicate]

We know that $J$ is an ideal of the full matrix ring $S=M_n(R)$ if and only if $J$ is the ring of all $n\times n$ matrices over $I$ for some ideal $I$ of the ring $R$ with identity. Now, my question ...