# Tagged Questions

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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### Can we build infinite products in $k[[X]]$?

Let $P \in k[[X]]$, where $k[[X]]$ denotes the ring of formal power series over the field $k$. Is well defined $$\prod_{n\in \mathbb{N}}P$$ (i.e. the power to infinity of $P$?) By looking at the ...
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### Any ring of prime order commutative ?

Is any ring of prime order commutative ?
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I am trying to show the following is a commutative ring with unity, however I am encountering a problem. First, addition and multiplication are defined as: $$a \oplus b=a+b-1$$$$a \odot b=ab-(a+b)... 1answer 29 views ### Prove that \forall a,b\in\mathbb{C} \exists c,d \in \mathbb{C}: c+d = a \land cd = b Prove that: \forall a,b\in\mathbb{C} \exists c,d \in \mathbb{C}: c+d = a \land cd = b We just learned about the characteristic/minimal polynomial and diagonalization but I am not sure if it has ... 1answer 61 views ### Example of non-commutative ring with exactly 2014 two sided-proper ideals. Find a non-commutative ring with exactly 2014 two sided-proper ideals. Find a ring with exactly 2014 pairwise non-isomorphic irreducible modules. If it was the commutative ring i would have thought ... 1answer 32 views ### Minimal right ideals Let I be a minimal right ideal of a ring R with 1. If r\in R, could we say that rI is zero or a minimal right ideal? I assumed a right ideal J in rI and intersecting it with I got a ... 1answer 90 views ### homomorphism between cohomology induced by the multiplication of an H-space Define the product on \mathbb{C}P^\infty in the following way: \begin{eqnarray*} \phi:\mathbb{C}P\overset{\Delta}\longrightarrow(\mathbb{C}P^\infty)^k\overset{\mu}\longrightarrow \mathbb{C}P^\infty,... 0answers 41 views ### Doubt related to quotients (group or ring) I was reading some notes about ring theory and modules and I've encountered with the following isomorphism: \mathbb (R[X]/ \langle x^3-1\rangle)/ \langle x-1\rangle \cong \mathbb R[X]/ \langle x-1 \... 1answer 24 views ### common factors of multilinear polynomial Say F,G\in\Bbb R[x_1,x_2,\dots,x_{n-1},x_n] are two multilinear polynomial. If F and G vanish at a common set of coordinantes (a_{i1},a_{i2},\dots,a_{in-1},a_{in})\in\Bbb R^n for i=1,\dots,t... 1answer 215 views ### If n\mid m prove that the canonical surjection \pi: \mathbb Z_m \rightarrow \mathbb Z_n is also surjective on units Not sure if this is the right proof (i found it online): Since n\mid m, if we factor m = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}, then n = p_1^{\beta_1}p_2^{\beta_2}\cdots p_k^{\... 2answers 41 views ### Computing the inverse of an element in \mathbb{Z}_5[x] / \langle x^2+x+2\rangle How does one calculate the inverse of (2x+3)+I in \mathbb{Z}_5[x] / \langle x^2+x+2\rangle? Give me some hint to solve this problem. Thanks in advance. 1answer 185 views ### algebra with topology homework problem Hello Everyone, I have this homework problem, I'm going to share what i have so far, not sure if Im in the right path. First, I have:$$f \sim g \, \Leftrightarrow \,x_0 \in \mathbb{R^n}, \exists \,U ...
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Questions: [See below] $\rm\color{#c00}{(1)}$ What structure has $R/I$ when $I$ is a maximal left ideal? I know that if $I$ is a bilateral ideal then $R/I$ is a ring and, moreover, if $I$ is ...
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### Counterexample for the infinitely many primes between two primes in a Noetherian ring

Consider the following Proposition: Proposition: Let $R$ be a noetherian ring. If $p_0 \subsetneq p_1 \subsetneq p_2$ is a chain of distinct prime ideals in $R$, then there exist infinitely many ...
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### Multiplicative group of a field contains maximal n-1 elements with order n

Let $F$ be a field and $n\in \mathbb N,n>1$. I want to show that the multiplicative group $K$\ $\{0\}$ contains maximal $n-1$ elements with order $n$. I actually don't have any ideas how to solve ...
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### A finite-dimensional $K$-algebra?

Let $A$ be a commutative noetherian domain of characteristic $p>0$ and $K$ be its quotient field. Let $G$ be any finite group whose order is divisible by $p$. Is $KG$ a finite-dimensional $K$-...
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### Looking for help, Aluffi Exercise 5.13, Chapter 6: characterization of PIDs

I quote: "Let $M$ be a finitely generated module over an integral domain $R$. Prove that if $R$ is a PID, then $M$ is torsion-free if and only if it is free. Prove that this property characterizes ...
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### $M_n(D)$ is left and right-simple?

Is it true that if $D$ is a division ring and $n\in\mathbb{Z}_{\geq1}$, then the only left and right ideals of the ring $M_n(D)$ are the trivial ones? I know that $M_n(D)$ is simple, and the proof ...
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### Proving that this mapping is one to one

Let $Q$ be the field of quotients of the Gaussian integers (integer complex numbers) and let $R$ be the the set of all complex numbers of the form $a +bi$ such that both $a,b$ are rationals I have to ...
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### Concerning Ideals and invertible elements in a commutative ring

Here is the problem that I have: Let $R$ be a commutative ring with unity and let $I$ be an ideal in $R$. Prove that $I=R$ if and only if $I$ contains some invertible element of the ring $R$. Here ...
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### If $f$ is a unit in a polynomial ring then $a_0$ is unit and all other coeficients are nilpotent.

I'm trying to prove the converse of the following theorem. I think suggestion available at this website are mistaken or I didn't understand them correctly. Theorem. Let $R$ be a commutative ring with ...
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### Quotient field of gaussian Integers

Let $D$ be the set of all gaussian integers in the from of $m+ni$ where $m,n \in Z$ Carry out the construction of the quotient field $Q$ for this integral domain.Show that this quotient field is ...
First of all, this is a homework problem. Let $C^{\infty}(\mathbb{R})$ denote the ring of smooth functions. Let $I_n$ denote the set of $f\in C^{\infty}(\mathbb{R})$ such that $$f^{(k)}(0)=0, \ 0 \... 1answer 91 views ### Why F[x]/p(x) would contain F? I am reading Abstract Algebra by Hungerford, and I am really confused about how we can extend a ring to a bigger ring. Here's what I got from the book: F be a field and p(x) be a nonconstant ... 1answer 87 views ### Proving the ring \mathbb{Q}[\mathbb{Z}] is not artinian My proposed solution: For each n \in \mathbb{N}, \mathbb{Q}[2^{n}\mathbb{Z}] is an ideal of \mathbb{Q}[\mathbb{Z}] (I think) and so we have the following infinite descending chain of ideals:... 2answers 71 views ### Matrix rings and ideals How would one go about checking if a given 2 x 2 matrix is an ideal. I am unclear as to what an ideal is and would like to know the steps in order to make the verification. Also, if it helps, I had ... 1answer 274 views ### I is an ideal in R implies that I[x] is an ideal in R[x]. Is the following statement right? If I is an ideal in the ring R, then I[x] is an ideal in the polynomial ring R[x]. If so, how can I prove it? 1answer 50 views ### Factoring isomorphism I have \mathbb Z[i\sqrt2] = {a+bi\sqrt2; a,b \in \mathbb Z, i^2=-1} and I =\{a+bi: a,b \in \mathbb Z, i^2=-1, 11\mid a+3b\}. My task was to prove that I is an ideal in \mathbb Z[i\sqrt2] by ... 1answer 52 views ### Showing certaing Integral domain is not well ordered. Let \mathbb{Z}[\sqrt2]=\{ a+b\sqrt2 ~| a,b\in \mathbb{Z}\}  be an integral domain. Let p=\{ a+b\sqrt2 ~|a,b\in\mathbb{Z} ~~~ and ~~~ a+b\sqrt 2 ~~is~~a~positive~real~number \}  Show that \... 2answers 441 views ### Is that Ring a field? Given a commutative Ring R of ordered pairs (x,y) of reals x,y with addition and multiplication defined in the following way.$$(x,y) + (u,v) = (x+u,y+v)(x,y).(u,v) = (xu-yv,xv + yu) I ...
If we define $(x,y).(u,v) = (xu - yv,xv + yu)$ do we have any non zero divisors for this meaning can we find non zero elements $(x,y)$ and $(u,v)$ such that $(x,y).(u,v) = (0,0)$ i tried to think of ...