This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

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1answer
7 views

Perfect-power Gaussian integer factorization

In $\mathbb Z[i]$, consider a relation $\alpha\beta=\epsilon\gamma^n$ for $\epsilon$ a unit and $(\alpha,\beta)=1$. Then why are each of $\alpha,\beta$ associated to nth powers $\xi^n,\eta^n$? ...
12
votes
1answer
59 views

Is the ring of holomorphic functions on $S^1$ Noetherian?

Let $S^1={\{ z \in \Bbb{C} : |z|=1 \}}$ be the unit circle. Let $R= \mathcal{H}(S^1)$ be the ring of holomorphic functions on $S^1$, i.e. the ring of functions $f: S^1 \longrightarrow \Bbb{C}$ which ...
1
vote
1answer
21 views

Generalization of a Result on Modular Inverses

Yesterday, I attempted to solve the general system of linear congruences (I'm not sure why I've never tried this before.) \begin{align*} x &\equiv a \pmod{A} \\ x &\equiv b ...
1
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0answers
22 views

tensor products of Hopf algebras

Let $H_1,\cdots, H_n$ be Hopf algebras over the field $\mathbb{Z}_p$, $p$ prime. Then the tensor product $$ \bigotimes_{i=1}^nH_i $$ is still an algebra over $\mathbb{Z}_p$. Is $ \otimes_{i=1}^nH_i $ ...
4
votes
2answers
53 views

Finding the kernel of maps between (polynomial) rings

If I have a map between rings like $f\colon k[x_1,x_2]\to k[t],x_1\mapsto t^2-1,x_2\mapsto t^3-t$, how can I prove that the kernel is $\mathfrak{a}=(x_2^2-x_1^2(x_1+1))$? I see that ...
3
votes
1answer
29 views

localization of the Pontrjagin ring of an $H$-space with respect to $\pi_0$

Let $X$ be an $H$-space. Let $F$ be a field. Then $H_*(X;F)$ is a Hopf algebra over $F$. According to group-like elements in the Hopf algebra of the homology of H-spaces, $$ \pi_0(X)=\{g\in ...
0
votes
2answers
65 views

Prove that an ideal $ \mathfrak{m} $ of a commutative ring $ R $ is maximal iff $ R/\mathfrak{m} $ is simple.

Could someone give me a hint on whether I’m on the right track or not? For sufficiency, I tried the following: Suppose that $ \mathfrak{m} $ is a maximal ideal. With the quotient map, we get $ ...
-2
votes
0answers
43 views

Chain of three prime ideals of $\mathbb F_p[x,y]$ [on hold]

Let $A_1\neq\left \{ 0 \right \} $, $A_2$, and $A_3$ be prime ideals of $\mathbb{F}_p[x,y]$ such that $$A_1\subset A_2\subseteq A_3\subset \mathbb F_p[x,y]$$ Then $A_2 = A_3$.
0
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1answer
41 views

What's a diagonal sum of two matrices?

Let $A$ be an $n\times n$ matrix over a field $K$. Show that there exists an invertible matrix $P$ such that $P^{-1}AP$ is a diagonal sum of an invertible matrix and a nilpotent matrix. (Hint: use ...
4
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2answers
63 views

“If $x$ is a non-unit, then $1-ux$ is a unit”

I don't understand these two lines from my book. We are given that $R$ is a local ring. If every $2$-generator submodule is cyclic and $Ra$, $Rb$ are given, then $Ra+Rb=Rc$, hence ...
0
votes
1answer
32 views

What's a $2$-generator submodule?

Context: Let $R$ be a local ring and $M$ an $R$-module. Show that the set of all submodules of $M$ is totally ordered by inclusion iff every finitely generated submodule of $M$ is cyclic or, ...
4
votes
1answer
37 views

group-like elements in the Hopf algebra of the homology of $H$-spaces

Let $X$ be an $H$-space with product $\mu$. Then $H_*(X)$ is a Hopf algebra with product $\mu_*$. Let $\psi$ be the coproduct of the Hopf algebra $H_*(X)$. Define a subset $S$ of $H_*(X)$ as $$ ...
3
votes
1answer
96 views

Subring of $M_7(\mathbb{Z}_2)$ isomorphic to $\mathbb{F}_{128}$?

Let $A \subset M_7(\mathbb{Z}_2)$ be a subring such that no proper nonzero subgroup $V \subset \mathbb{Z}_2^7$ is invariant under all matrices in $A$. I suspect that $A \cong \mathbb{F}_{128}$, but ...
0
votes
1answer
12 views

solution verification: find characteristic of integral domain under given conditions

Okay, so this seems an easy problem, but I was having doubts if my solution was correct or not. I would really appreciate if somebody could verify it for me. Suppose $R$ is an integral domain such ...
1
vote
0answers
25 views

Conjecture concerning involutions in a unitary braided fusion category/Grothendieck ring

Despite the categorical setup, a solution to this question may require no categorical tools (see Conjecture 2). Let $\mathcal C$ be a unitary braided fusion category, $I$ be its set of isomorphism ...
1
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3answers
71 views

What does $\overline{r}m:=rm$ mean?

On this Wikipedia article, it says that you can define an $R$-module $M$ as an $R/Ann_R(M)$-module using the action $\overline{r}m:=rm.$ What does that action actually mean? What is $\overline{r}$?
1
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0answers
22 views

Decompose finitely generated modules and use Krull-Schmidt theorem [duplicate]

I'm trying to show that if $R$ is an Artinian ring, then for finitely generated modules $M,N,N'$, we have that $M\oplus N\cong M\oplus N'$ implies that $N\cong N'$. I'm supposed to do this by ...
1
vote
2answers
71 views
+50

Modules that have finitely many submodules

Drawing the lattice of submodules of a given module helps me to gain some intuition about the structure of module. Sometimes, however, it is not possible to draw in neat manner; For example vector ...
3
votes
2answers
50 views

Constructing DVR's from arbitrary UFD's

Is the following statement true? Let $A$ be an UFD and $p\in A$ prime, then $A_{(p)}$ is a discrete valuation ring. I think yes: For every element $x$ of $Q(A_{(p)})=Q(A)$, there is a unique ...
3
votes
0answers
40 views

What properties do the rings of infinite, upper-triangular matrices have?

I'm very familiar with the ring of $n\times n$ matrices over a field, the ring of $n\times n$ upper triangular matrices over a field, and the ring of infinite column-finite matrices over a field. But ...
7
votes
1answer
93 views

different definitions of Hopf algebras

(i). In the book Algebraic Topology, A. Hatcher, p. 283, the notion Hopf algebra is defined as follows: (ii). However, in the book Bialgebras and Hopf algebras, J.P. May, the notion Hopf algebra is ...
1
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0answers
46 views

Nilpotent minimal ideals are one sided minimal ideals

Let $R$ be an Artinian ring. I am looking for some condition under which all nilpotent (two sided) minimal ideals of $R$ are both minimal left and minimal right ideals. Can anyone give some hint? ...
4
votes
1answer
94 views

When $N \to M \otimes_R N$ is not an embedding.

Can someone please provide an example of the following (or tell me why such an example doesn't exist): Let $R$ be a (not necessarily commutative) ring, $M$ an $R$-$R$-bimodule containing a copy of ...
0
votes
1answer
33 views

Ideals for non commutative ring

For a non commutative ring without identity, is it possible that there will be right and left ideals which are different?
2
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3answers
32 views

Finding Linear Combination of Polynomials

I am stuck on a question involving finding the greatest common divisor of polynomials and then solving to find the linear combination of them yielding the greatest common divisor. My work thus far is ...
2
votes
1answer
37 views

Why do we need injectivity in the definition of integral dependence?

Let $f: A \rightarrow B$ a ring morphism of commutative rings, then one has on $B$ a multiplication by elements of $A$ defined by $b*a \doteq b.f(a)$ (where . is the multiplication in the ring $B$). ...
0
votes
1answer
63 views
2
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2answers
52 views

Ring Theory: Identity Elements

A brief question$:$ If a ring is specified to have an identity, is it implicit that the identity in question is the multiplicative identity? From the definition of a ring $R$, $R$ must contain ...
1
vote
1answer
18 views

Show that in ascending Loewy series, $S^r(R)=R$

Let $R$ be an Artinian ring, $N$ its radical, and $r$ the smallest natural number such that $N^r=0$. Define an ideal $S^n(R)$ of $R$ recursively as follows: $S^1(R)=soc(R)$ Assuming ...
7
votes
1answer
71 views

Proving Things About Rings Using Things About Vector Spaces

All rings below are assumed to be commutative and having an identity. $\newcommand{\bw}{\bigwedge}\newcommand{\R}{\mathbf R}\newcommand{\mc}{\mathcal}$ Consider the following problem: Problem 1. ...
1
vote
1answer
39 views

Understanding this proof regarding maximum ideals

Prove: Let $R$ be a commutative associative ring with $1$ and $M \triangleleft R$. The factor-ring $R/M$ is a field iff $M$ is the max ideal in $R$. ($\implies$) Let $R/M$ be a field, then ...
4
votes
1answer
67 views

Generalization of the derivative to polynomial rings

It is easy to see why the derivative plays an important role in real and complex analysis from the geometric viewpoint. However, one can extend the definition of a derivative to polynomial rings such ...
0
votes
1answer
29 views

Is the ideal $\{2m + (1 + \sqrt{-6})n:m, n\in\mathbb{Z}\}$ principal in $\mathbb{Z}[\sqrt{-6}]$?

Is the ideal $\{2m + (1 + \sqrt{-6})n:m, n\in\mathbb{Z}\}$ principal in $\mathbb{Z}[\sqrt{-6}]$? I have an exercise that asks just that. As a hint it says to prove that this ideal contains $1$, ...
4
votes
0answers
68 views

What properties $R \subseteq S$ should have in order that every prime ideal of $S$ is extended?

My question is almost the same as In what conditions every ideal is an extension ideal?; I allow myself to ask this question, since there is no answer to the above question. My question: Given ...
2
votes
2answers
35 views

Equivalent definitions of faithful flatness

I am currently studying the equivalent definitions of faithful flatness over (probably noncommutative) unital rings. In particular, there is a version that I have doubts about: Let $R \subseteq S$ ...
3
votes
1answer
61 views

Intuition behind $k$-algebra, $k$-algebra morphisms?

I will state the definition of a $k$-algebra and $k$-algebra morphisms. A ring $A$ equipped with a ring homomorphism $k \to Z(A)$ is called a $k$-algebra. More explicitly, this means that $A$ has ...
2
votes
1answer
33 views

A question about 1.0.3 in Grothendieck's EGA

In (1.0.3), Grothendieck states that, given non-commutative rings $A$ and $B$, a homomorphism $\varphi : A \to B$, and a left ideal $\mathfrak{J}$ of $A$, the left ideal $B\mathfrak{J}$ of $B$ ...
1
vote
6answers
72 views

Constructing $\mathbb{C}$ from $\mathbb{R}$

I'm having difficulty grasping the notion that you can define the complex numbers as $\mathbb{C}=\mathbb{R}[t]/\langle t^2+1\rangle$. As far as I understand, $\mathbb{R}[t]$ is the set of all ...
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0answers
40 views

Example of indecomposable self injective ring [closed]

Please give an example of a commutative indecomposable self injective ring with 4 or more maximal ideals.
2
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1answer
37 views

If $R$ is a simple ring, is every corner $eRe$ simple?

Assume that $R$ is a ring, not necessarily commutative nor unital. Let $e$ be an idempotent in $R$. Is there a correspondence between ideals of $R$ and ideals of the corner ring $eRe = \{ere : r\in ...
3
votes
1answer
76 views

If $A$ is a finitely generated $R$-module, is $\operatorname{Hom}_R(A,R)$ finitely generated? [duplicate]

Let $R$ be an utterly arbitrary commutative, unital ring. Let $A$ be a finitely generated $R$-module. Is $\operatorname{Hom}_R(A,R)$ finitely generated as an $R$-module? Intuitively and based on ...
0
votes
1answer
56 views

Module of constant rank over noetherian reduced ring

Let $A$ be a reduced noetherian commutative ring and $M$ be a finitely-generated $A$-module such that for all prime ideals $\mathfrak p$, $M_{\mathfrak p}/\mathfrak pM_{\mathfrak p}$ is an ...
1
vote
2answers
60 views

Why do nil ideals annihilate simple modules?

A nil ideal $N$ of a ring $R$ is defined as follows: $(N,+)$ is a subgroup of $(R,+)$ $\forall x \in N, \forall r \in R :\quad x \cdot r \in N$ $\forall x \in N, \forall r \in R : \quad r \cdot x ...
0
votes
0answers
22 views

Looking for a Coordinate Free Way to Prove a Precursor to Nakayama Lemma.

Let $M$ be a finitely generated module over a ring (commutative with identity) $R$. Let $\mathfrak a$ be an ideal of $R$ and $\phi:M\to M$ be an $R$-module homomorphism such that ...
4
votes
0answers
37 views

Too Many Members in a Finitely Generated Module are Linearly Dependent

I am new to module theory and as of now am not very comfortable with the subject. So can somebody please check whether my claim and its proof is okay? Consider the following statement: Let $M$ be ...
0
votes
1answer
33 views

If $F$ is a field, then any two algebraic closures are isomorphic by an isomorphism that is the identity on $F$.

To start, suppose $K_1$ and $K_2$ are two algebraic closures of $F$. (a) Let $P$ be the set of partial functions $f$ from $K_1$ to $K_2$ with the following properties: $F$ is contained in ...
0
votes
1answer
25 views

Do real quadratic fields with unique primary factorization exist?

Bumped in Stillwell's book "Elements of Number Theory" into "The real quadratic fields with unique prime factorization are still not known ...". But doesn't $\mathbb{Q}[\sqrt{2}]$'s ring of integers ...
1
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1answer
25 views

Prove socle is ideal

In any ring $R$ define the socle as the sum of all minimal right ideals of $R$. Say we have two minimal ideals $A,B$. If $a\in A,b\in B$, then $a+b$ is in the socle. If $x\in R$, then ...
1
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3answers
28 views

Centre of matrix ring over skew field

Let $R$ be a semisimple ring. Show that $R$ is simple iff the centre of $R$ is a field. Book's solution: If $R$ is simple, it has the form $\mathfrak{M}_n(K)$ for a skew field $K$, and its ...
-1
votes
1answer
42 views

Find the number of units in $M_2(\mathbb Z_p)$ where $p$ is prime. [duplicate]

Find the number of units in $M_2(\mathbb Z_p)$ where $p$ is prime. I want a detailed solution, not just the number. $M_2$ means matrix of order $2\times 2$. I know the defn of units. But how to ...