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

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2answers
44 views

Example of the semisimple ring $R$ but $R^{{\rm op}}$ is not.

Is there any example of this kind of rings? i don't have any imagine of this rings, if they are exist!
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0answers
20 views

Difficulties to understand the Rowen's notations

Some definitions... Definition 1: A polynomial $f(X_1,\dots ,X_d)$ is $t$-linear if the variables $X_1,\dots ,X_t,\; t\leq d$ appear in all monomials of $f$ and degree of $X_i,\; i=1,2,\dots ,t$ on ...
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1answer
42 views

How do I find the spectrum of a ring?

What is $Spec R$ where $R$ is the integers modulo $6$? More generally, what are the techniques to find the spectrum of any commutative ring? (I would also be interested in the non-commutative case but ...
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1answer
48 views

Prime which is not Irreducible in ring with unity without zero divisors (not necessarily commutative)

In a non-commutative ring with unity without zero divisors find a prime element which is not irreducible (if possible). $p$ is prime iff $p|ab$ implies that $p|a$ or $p|b$, and $x$ is irreducible iff ...
2
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1answer
87 views

Non-commutative ring (not necessarily with multiplicative identity) of order $n$ exists if and only if $p^2|n$ for some prime $p$?

Is it true that there is a non-commutative ring (not necessarily with unity) of order $n$ if and only if $p^2\mid n$ for some prime $p$ ?
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0answers
16 views

Tensor product and opposite rings

Suppose $A$ is a ring, $M$ is a right $A$-module, and $N$ is a left $A$-module. In this situation, we can form the tensor product, $M\otimes_A N$, and this is an abelian group (and even a ...
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0answers
18 views

A question on a decomposed projective module

Let $R$ be a right semi-perfect ring and $P$ a projective right $R$-module. It is a well-known fact that $P$ is isomorphic with a direct sum $(e_1R)^{(A_1)}⊕...⊕(e_nR)^{(A_n)}$, where $e_1...e_n$ is a ...
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1answer
36 views

Prime elements in a noncommutative ring

Is there a reasonable definition of prime element in a noncommutative ring? The definition from wikipedia makes the assumption of commutativity and I'd like to know how necessary this condition is. ...
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2answers
67 views

Showing minimal graded free resolutions are isomorphic

I'm currently reading Rogalski's notes on noncommutative projective algebraic geometry (which can be found here) and I'm currently trying to fill out the details of Lemma 1.24 (2). The step which I ...
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0answers
14 views

Projectivity of a certain module

Let $M$ be a generator for the category of left $R$-modules, and let we have an $R$-epimorphism $h$ from $R^{(X)}$ to an $R$-module $P$ which is projective relative to $M^{(X)}$ ($X$ is a set). I want ...
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0answers
52 views

On the group algebra of a group specified by generators and relations

Let $k$ be a field, $G = \langle S \rangle/N(R)$ be a group specified by a set of generators $S$ and a set of relations $R$ (the brackets denote the free group, and $N(R)$ means the conjugate closure ...
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0answers
9 views

Generators Precisely Cogenerators

I want to prove that the following are equivalent: 1) a ring $R$ is simple artinian; 2) each nonzero left $R$-module is a generator; 3) each nonzero left $R$-module is a cogenerator. My try: if ...
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2answers
42 views

$K$-basis of cyclic $KG$-module determined by group elements

Let $G$ be a finite group and $K$ a field with $\mathrm{char}(K) \nmid |G|$, and let $M$ be a cyclic $KG$-module. For a cyclic generator $v \in M$ we can pick group elements $g_1, \dots, g_d \in G$ ...
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1answer
23 views

A question on decomposition of modules

Let $M$ be an $R$-module. A decomposition $M=⊕_A M_\alpha$ of nonzero submodules is said to complement direct summands in case for each direct summand $K$ of $M$ there is a subset $B$ of $A$ with ...
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1answer
31 views

A question about $rad(M)$

We know that for any $R$-module $M$ the radical $rad(M)$ is the sum of all small submodules of $M$. My question: "if $x\in rad(M)$ is it true in general that $Rx$ is a small submodule of $M$? ...
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0answers
32 views

Superfluous radical

If $M$ is a projective Artinian (left) module over a ring $R$, could one say that the radical of $M$ is a superfluous submodule in M? I know that for a projective module $M$ the radical of $M$ equals ...
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1answer
26 views

Projective modules over enveloping algebra

How could I prove the following statement? Let $k$ a commutative ring and let $A$ an associative $k$-algebra that is also a projective $k$-module. Then every projective $A\otimes A^{op}$ left module ...
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1answer
9 views

Associative and anticommutative Binary Operation(composition)

Show that if binary operation ,$\Delta$, is associative and anticommutative on $\mathbb{E}$, then $x\Delta y \Delta z=x\Delta z$ ∀$x,y,z \in \mathbb{E}$. [Hint: consider $x\Delta y\Delta z\Delta ...
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0answers
20 views

Conditions equivalent to Noetherianness

Let $R$ be a left Noetherian ring. We know that any direct sum of injective left $R$-modules is again injective. Since any injective module is quasi-injective, we infer that (1):"any direct sum ...
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0answers
26 views

Nilpotent Jacobson radical of $End(M)$

If $M$ is a Noetherian injective left $R$-module, is it true that the Jacobson radical $J=J(End(M))$ of the endomorphism ring of $M$ is nilpotent? I know that if a left $R$-module is Artinian or ...
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1answer
49 views

A free direct sum of a projective module

I want to prove that a left module $_RP$ is a projective generator if and only if a direct sum of (copies of) $P$ is free. My try is, first, to observe that $P$ is a generator if and only if for ...
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2answers
31 views

Are dual numbers a special case of grassmann numbers?

Dual numbers are defined in analogy to complex numbers like $$ z = a + \varepsilon b. $$ But instead of $i^2=-1$ it is defined that $\varepsilon^2=0$. The multiplication rule for Grassmann numbers ...
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0answers
39 views

Simple composition factor

We know that if $e$ is a primitive idempotent in a semiperfect ring then $Re/Je$ is a simple left $R$-module, where $J$ is the Jacobson radical of $R$. Now, suppose that $e$ is an idempotent in an ...
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1answer
31 views

Generators over semiperfect rings

It is clear that if $R$ is a ring with identity and $e\in R$ is an idempotent then $Re$ is a direct summand of $R$ while $R$ is a generator in the category of left $R$-modules. I have my question when ...
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4answers
84 views

An elementary fact about tensor product of modules

Let $A$ be an $R$-right module, $N$ be a submodule of $R$-left module $M$, $\pi:M\rightarrow M/N $ is the natural epimorphism. What is $\ker\left(\pi\otimes1_{A}\right) $? I appreciate your help.
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0answers
68 views

(Soft Question) How active an area of research is Non-Commutative Geometry? [closed]

I am currently an undergraduate, but I am considering applying for a phd in algebraic geometry or a related field. I am quite interested in the link between non-commutative geometry and theoretical ...
6
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1answer
37 views

the Zassenhaus /Baker–Campbell–Hausdorff formula for cosine.

This question concerns the expansion of non-commutative algebra $[X,Y] \neq 0$ for two operators $X,Y$. One can think of $X$ and $Y$ as some matrices. If $[X,Y] = 0$, we have $$e^{t(X+Y)}= e^{tX}~ ...
0
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1answer
38 views

Simple $R$-modules are isomorphic [duplicate]

Let $F$ is a field, and $R = M_n(F)$. Prove that (i) Every minimal left ideal of $R$ are isomorphic to each other (ii) Prove every simple $R$-module are isomorphic to each other I have if $M$ is a ...
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0answers
59 views

Noncommutative rings - binomial expansion

Let $R$ be a noncommutative ring, and $a,b\in R$. We can define $a^{(0)}=a, \ a^{(1)}=[a,b]$ and $a^{(k)}=[a^{(k-1)},b]$. Show that ...
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2answers
253 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 ...
3
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1answer
30 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')\\ ...
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0answers
15 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 ...
3
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1answer
49 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, ...
1
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1answer
24 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
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1answer
93 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
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2answers
49 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 ...
2
votes
2answers
52 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 ...
3
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2answers
146 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 ...
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0answers
39 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, ...
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0answers
24 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
32 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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0answers
33 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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0answers
11 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
42 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 ...
3
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1answer
73 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
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0answers
26 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
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1answer
27 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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0answers
36 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
49 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} ...
1
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1answer
31 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 ?