Tagged Questions

6
votes
0answers
64 views

Amenable group rings embeddable in skew fields

I'm looking for a reference of the following fact: given a (countable?) amenable group $G$ and a (skew) field $K$, the following are equivalent: (1) the group ring $K[G]$ is a domain; (2) $K[G]$ is ...
0
votes
0answers
44 views

Open Problem in Division Ring

There exists prominent problem in division ring theory Let $D^*$ be finitely generated division ring. Then $D$ is finite. Does exist a survey about this problem? Or papers?
3
votes
1answer
26 views

Reference request: Morita contexts

During an independent study I've come across Morita contexts, but I'd like to understand them better. A quick Google search doesn't yield much fruit, so I was hoping to find a good reference on the ...
4
votes
0answers
25 views

An explicit $\Lambda_R^\ell(M)$ when $M$ is not free

Let $\Lambda_R^n(M)$ be the nth exterior power of an $R$-module $M$. Let us assume $M$ is finitely generated. When $M$ if free, say, $M=R^{\oplus d}$, we know \begin{equation} \Lambda_R^n(M)\cong ...
3
votes
0answers
59 views

When is $\mathbb{Z}\Gamma$ a left Noetherian ring?

Denote $\Gamma$ to be a countable discrete group, let $\mathbb{Z}\Gamma$ to be its integer group ring. A ring is left-Noetherian if it satisfies the ascending chain condition on left ideals.(c.f. ...
3
votes
2answers
52 views

Subring of $R$ with fraction field $=$ Frac $R$

Let $R$ be an integral domain with fraction field $K$. Of course all the overrings of $R$ share the same fraction field $K$ (by an overring I mean a subring $S\subset K$ containing $R$ as a subring). ...
5
votes
6answers
251 views

Abstract algebra book recommendations for beginners.

I am taking abstract algebra in this semester. I find that my lecturer didn't provide enough examples for understanding. So I would like to ask what are the recommended books which has lots of solved ...
2
votes
2answers
53 views

Why is $S/R$ a ring extension?

If $S$ is a ring and $R \subset S$ is a subring it's common to write that $S/R$ is an extension of rings. I frequently find myself writing this and read it quite often in textbooks and lecture notes. ...
4
votes
2answers
129 views

Epic maps in the category of commutative rings with identity.

Here all rings are assumed to lie in the category $\cal C$ of commutative rings with identity, and ${\cal C} (\ R\ ,\ S\ )$ is the set of all ring homomorphisms $F$ from $R$ to $S$ for which ...
7
votes
1answer
119 views

The Picard-Brauer short exact sequence

It seems to be a rather well understood fact that, given commutative rings $R,S$, and a homomorphism $R \to S$ there is a short exact sequence $$\text{Pic}(R) \to \text{Pic}(S) \to F_0 \to ...
4
votes
1answer
113 views

How can I calculate the radical of an ideal in ring ${\Bbb Z}_n$?

I learned the concept radical of an ideal from this wikipedia article. I tried some examples and I found that it's not easy to find $Rad(I)$. (That article gives some examples when $R={\Bbb Z}$.) For ...
3
votes
1answer
110 views

Direct limits and $\rm Hom$

I read that $\lim\limits_{\longleftarrow}\mathrm{Hom}(N_j,M)\cong\mathrm{Hom}(\lim\limits_{\longrightarrow}N_j,M)$. I was wondering if we can write $\lim\limits_{\longrightarrow}\mathrm{Hom}(N_j,M)$ ...
6
votes
4answers
199 views

Spectrum of $\mathbb{Z}[x]$

Can someone point me towards a resource that proves that the spectrum of $\mathbb{Z}[x]$ consists of ideals $(p,f)$ where $p$ prime or zero and $f$ irred mod $p$? In particular I remember this can be ...
5
votes
1answer
165 views

Polynomials in matrices with integer entries

I'm looking for references, if there is any, for this problem: Characterize all elements $a \in M_n(\mathbb{Z})$ for which we have $\mathbb{Q}[a] \cap M_n(\mathbb{Z})=\mathbb{Z}[a].$ Here, by ...
2
votes
1answer
141 views

Relaxing the definition of a von Neumann regular ring

Hereinafter, all rings are assumed to be unital but not necessarily commutative. A well-known class of rings are von Neumann regular rings, that is, rings $R$ such that for each $a\in R$ there is an ...
3
votes
1answer
101 views

Theorem of Kaplansky, $R$ is a division ring if every element but one is (right) quasi-invertible.

There is a theorem of Kaplansky that seems to pop up every algebra book. Here rng denotes a ring with possibly no identity. As definition, an element $a$ of a rng $R$ is said to be (right) ...
1
vote
0answers
34 views

Why do we have $\operatorname{Proj}(\oplus_{i=0}^\infty t^i \left< x\right>^i)\cong \operatorname{Proj}(\oplus_{i=0}^\infty t^i \left< x^2\right>^i)$?

I'm just curious but why is it that $$ \operatorname{Proj}\left(\oplus_{i=0}^\infty t^i \left< x\right>^i\right) $$ isomorphic to $$ \operatorname{Proj} \left(\oplus_{i=0}^\infty t^i \left< ...
9
votes
1answer
200 views

Why is the absence of zero divisors not sufficient for a field of fractions to exist?

I've recently begun to read Skew Fields: The General Theory of Division Rings by Paul Cohn. On page 9 he writes, Let us now pass to the non-commutative case. The absence of zero-divisors is still ...
8
votes
1answer
127 views

Ties between Lie algebras and ring theory

I would like to get a general understanding of the relationship between (noncommutative) ring theory and Lie algebra theory. All Lie algebras are finite dimensional and over a field $k$ of ...
3
votes
3answers
93 views

Are the $\mathbb{Z}/p^k\mathbb{Z}$ simple modules all isomorphic to one another?

After looking back over some finite field theory, I've been thinking about the ring $\mathbb{Z}/p^k\mathbb{Z}$ for some prime $p$. I'm just curious, are the $\mathbb{Z}/p^k\mathbb{Z}$ simple modules ...
4
votes
1answer
212 views

Ideals in non-associative rings and the identity $(xy)z=y(zx)$.

I have come across this paper. The authors prove that magmas satisfying the identity $$(xy)z=y(zx)\tag1$$ are nearly both associative and commutative. To be precise, they show that in such magmas, ...
4
votes
1answer
158 views

Ideal as a projective module

I am sorry, this may not be a good question here. I am looking a good reference about when the ideal I of a given commutative ring R (local or may not be local) with identity is a projective module.
0
votes
1answer
80 views

Do these ideals have names?

Given a ring $R$ and an ideal(two-sided) $I\subset R$, we find an ideal $$[R:I]=\{x\in R| xR\subset I \}$$ It is easy to see that this ideal coincides with the original ideal $I$ if $I$ is a prime ...
8
votes
2answers
361 views

What is an element of a rng called which is not the product of any elements?

Let $R$ be a non-unital ring. Let $F:R\times R\longrightarrow R$ be a function given by the formula $F(x,y)=xy.$ Let $r\not\in\operatorname{im}(F).$ Such elements can exists, for example $2\in ...
3
votes
3answers
506 views

Ring theory exercises at the graduate level

Do you know any book or an online source that contains exercises on ring theory? I've solved some exercises of Lang's Algebra and Dummit & Foote's Abstract Algebra but there is a huge gap between ...
2
votes
0answers
133 views

Linearly independent rows in a square matrix

Suppose $m,n \in \mathbf{N}, m\le n$. Let $A$ be a matrix with $\mathbf{Q}$ linearly independent $b_{1},...,b_{m}$ in $\mathbf{Z}^{n}$. a) Show that there are $v_{1},...,v_{m} \in \mathbf{N}$ so ...
6
votes
2answers
152 views

Example of a commutative perfect ring that is not artinian

I read a result here stating that a commutative perfect ring is artinian if and only if it is $(1,1)$-coherent (see Proposition 5.3). I'm interested in finding an example of a commutative perfect ...
5
votes
1answer
265 views

The only two rational values for cosine and their connection to the Kummer Rings

I am trying to understand Kummer Rings, and in particular what makes $n=3,4,6$ so special. (That is the Gaussian and Eisenstein integers) The norm of $\mathbb{Z}[\zeta_n]$ is ...
4
votes
1answer
181 views

Lifting maps of quotient modules

Today I tried to check this, but couldn't see how to do it. I think it is probably a standard result, but a brief check of Atiyah-Macdonald didn't yield anything, and I don't know what to google for. ...
2
votes
1answer
253 views

Name of a book with the following contents?

Some time ago, I received Algebra notes from my advisor who is advising me on a project. I learned very much from these notes and wondered what the name of the book from which these notes were ...
9
votes
2answers
350 views

Do people study “ring presentations”? Is this a dumb question?

So one way to define a group presentation is to say, well, let's generate the free group with some number of generators, and then quotient by saying certain elements (relators) cancel just as ...