0
votes
0answers
39 views

Ring $R$ as $R[x]$-module

My professor mentioned some interesting examples of modules, giving as an example the following two: $R$ as an $R[x]$-module, in which multiplication by $x$ was taken to be evaluation under a fixed ...
0
votes
0answers
23 views

Theory of irrationalities- Faddeev's book

Does anyone know where (if available) I can get a free access to Delone, B. N., Faddeev, D. K., ''The theory of irrationalities of the third degree'' Transl. Math. Monographs 10, Amer. Math. Soc., ...
1
vote
0answers
26 views

integral basis for an arbitrary cubic Galois field

I wonder where I can find some information (possible a book) about finding an integral basis for cubic Galois fields? I know that for pure cubic fields there exists a simple criterion according to ...
5
votes
1answer
68 views

The infinite Direct Sum in the category Ring

If you don't have strong personal feelings about it already, most of you have at least witnessed the opposing factions on how we should define a ring and, by extension, how we should define a ring ...
3
votes
1answer
94 views

Counterexample for $A[[x, y]] = A[[x]][[y]]$

Maybe this is an idiot question, but I've heard that $A[[x, y]] = A[[x]][[y]]$ does not hold for $A$ an arbitrary commutative ring with identity, so I would like to know a counterexample, since the ...
1
vote
1answer
119 views

How should I understand $R[x]/(f)$ for a ring $R$?

The following is a proposition in Artin's Algebra: Proposition 11.5.5 Let $R$ be a ring, and let $f(x)$ be a monic polynomial of positive degree $n$ with coeeficients in $R$. Let $R[\alpha]$ ...
2
votes
2answers
54 views

when is a ring a free module over a subring?

Let $S \subset R$ be rings, $S$ not necessarily an ideal of $R$, and $S \neq R$. Is there anything that can be said about when $R$ is free as an $S$-module?
0
votes
0answers
50 views

When is a vector space (over field $K$) also a ring (with subring $K$)?

(Apologies in advance for the very naive question. I'm just learning about all this. Also, for the sake of expedience, below I use the word "ring" when it would more correct for me to use ...
6
votes
1answer
78 views

Minimal counterexamples of the isomorphism problem for integral group rings

The isomorphism problem for integral group rings asks if two finite groups $G,H$ are isomorphic when their integral group rings $\mathbb{Z}[G]$, $\mathbb{Z}[H]$ are isomorphic. Quite a lot has been ...
13
votes
5answers
382 views

Introduction to ring theory?

I've been teaching myself algebra these couple of months. I already went through the basics of group (Lagrange, action, class equation, Cauchy and Sylow theorems etc.) And I already have some linear ...
1
vote
0answers
21 views

Two ordered rings are isomorphic iff their positive semirings are isomorphic

I am looking for (a reference to) a proof that Two ordered rings are isomorphic iff their positive ordered semirings are isomorphic. The positive semiring of an ordered ring $R$ is here the ...
6
votes
2answers
277 views

Proof of Fermats Last Theorem for Given Exponent

Where can I find reasonably short and elementary proofs (using basic concepts of ring, field, galois theory) of Fermat's Last Theorem for specific n? For example, $n=5,7,13$?
1
vote
2answers
83 views

References in Ring Theory

I am very interested in ring theory. I want to know the best textbooks about that subject, i.e about PID's, UFD's, Dedekind domain, Euclid domain. Can anyone tell me?
1
vote
1answer
36 views

Lemma of Bézout

Let $A$ be a PID. By the Lemma of B├ęzout I mean the statement that for elements $a_1,\ldots,a_n\in A$ we have $(a_1,\ldots,a_n)=((a_1,\ldots,a_n))$ where $(a_1,\ldots,a_n)$ denotes a greatest common ...
1
vote
1answer
28 views

proof of equivalent statements for an element of a ring belonging to every maximal left ideal of that ring

I would like to see a proof of the following. Let $R$ be a ring and let $a\in R$. Prove that the following conditions are equivalent. $a$ belongs to every maximal left ideal of $R$. $1+ra$ has ...
3
votes
0answers
57 views

Morita theory for simplicial rings

My question is the following: is there an analog of Morita theorem in the simplicial setting? I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories ...
2
votes
1answer
131 views

Finding Idempotents?

I was wondering if there is a method for finding primitive idempotents of a finite dimensional algebra (over a field)? or in other words is there any way to build the complete set of primitive ...
2
votes
2answers
54 views

A non-UFD where we have different lengths of irreducible factorizations?

A unique factorization domain (UFD) is defined to be an integral domain where each element other than units and zero has a factorization into irreducibles, and if we have two such factorizations then ...
3
votes
0answers
273 views

Proof that $\mathbb Z[\sqrt{3}]$ is Euclidean

Let $R_d$ be the ring defined as $R_d=\left \{ x+y\omega : x,y\in \mathbb{Z} \right\}$, where $$\omega = \begin{cases} \sqrt{d}, & \text{if } \quad d \not \equiv 1\mod 4 \\ \frac{1+\sqrt{d}}{2}, ...
2
votes
2answers
77 views

What topic/subject is this?

We can take a value $j$ and say $j^m = -1$. Now ordinarily, for the complexes, we could simply say that $j = e^{2 \pi i / m}$. I'm wondering, though, if there is a way to create a number system ...
5
votes
0answers
40 views

When are infinite dimensional path algebras hereditary

The title says mostly everything. Suppose we have a quiver, maybe with relations and cycles. Is it known when the path algebra modulo relations is hereditary. Especially in the case that the path ...
6
votes
0answers
189 views

Generalizing the natural numbers - has this been studied before?

The ordinals numbers generalize the natural numbers and satisfy a generalized induction principle. However, the algebraic properties of the ordinal numbers aren't so good. For example, ordinal sums ...
2
votes
0answers
66 views

Kronecker's approach to unique factorisation in algebraic number theory: books and references

I have done a short course (one semester) on algebraic number theory at beginning graduate level, in which the Dedekind theory of ideals features prominently. However I have since discovered that ...
0
votes
0answers
45 views

Ring structure on finite string of elements of a group

This is a reference request. Suppose $(G, \cdot)$ is a group and consider the structure on $G^{<\omega}$ where, for $\mathbf{p} = (p_1, \dots, p_n) \in G^{<\omega}$ and $\mathbf{q} = (q_1, ...
2
votes
0answers
60 views

A criterion for finite generation of subalgebras of a polynomial ring

In her 1926 paper on invariant theory, Emmy Noether uses a certain "finiteness criterion" which I wish to translate and maybe find a more modern reference to. The original wording is: Ein ...
16
votes
1answer
185 views

How 'commutative' can a non-commutative ring be?

Let $R$ be a finite non-commutative ring. Let $P(R)$ be the probability that two elements chosen uniformly at random commute with each other. Consider the value $$S=\sup_RP(R)$$ where the supremum ...
8
votes
1answer
108 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 ...
4
votes
1answer
44 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
26 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 ...
5
votes
0answers
151 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
81 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
834 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
66 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
143 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
165 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
201 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 ...
2
votes
1answer
237 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 $\lim\limits_{\longrightarrow}\mathrm{Hom}(N_j,M)$ ...
7
votes
4answers
328 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
203 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
197 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
166 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
35 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
264 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
180 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
108 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 ...
5
votes
1answer
283 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, ...
7
votes
2answers
317 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 ...
0
votes
1answer
83 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
381 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
726 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 ...