0
votes
2answers
50 views

Maximal ideal which isn't principal

Let $J=(x-2,x-y^2-3)$ ideal in the polynomial ring $\Bbb R[x,y]$. Please help me prove that $J$ is a maximal ideal which isn't principal, and that $\Bbb R[x,y]/J \cong \Bbb C$.
1
vote
0answers
62 views

Ideals of $\Bbb Z/p^2q\Bbb Z$

Let $p,q$ be distinct primes. Then $\mathbb{Z}/p^2q\mathbb{Z}$ has 3 distinct ideals. $\mathbb{Z}/p^2q\mathbb{Z}$ has 3 distinct prime ideals. $\mathbb{Z}/p^2q\mathbb{Z}$ has 2 distinct prime ...
2
votes
1answer
20 views

Show that if $R$ is principal, $N $ is pure and $Ann(x+N)= Rd$ then there exists $y \in M$ such that $x+N=y+N$ and $Ann(y)=Rd$

Let $R$ a integral domain and $M$ a $R$-module. A submodule $N$ of $M$ is pure if for all $x \in M$ and $a \in R$ such that $ax \in N$, there exists $y \in N$ such that $ax=ay$. Show that if $R$ ...
2
votes
1answer
29 views

Let $R$ a ring with maximum common divisor. If $a,b,c \in R$ such that $a|bc$ and $(a,b)=1$ then $a|c$.

Let $R$ a ring with maximum common divisor. Show that if $a,b,c \in R$ such that $a|bc$ and $(a,b)=1$ then $a|c$. Comments: I tried to use the Bezout's theorem, but in my course we saw it only ...
-1
votes
1answer
42 views

Ring of linear transformations modulo finite rank transformations [closed]

Let $ K $ be a field and $ V $ be a vector space of countable dimension (infinite) over $ K $, and let $ L = L (V) $ be the vector space of $ K $-linear transformations on $ V $. Let $ I $ be the ...
1
vote
1answer
25 views

Show that if $M$ is a R-module free, for all $r \in R$ and $m \in M$ such that $r \neq 0$ e $m \neq 0$ we have $rm \neq 0$.

is it true that if $M$ is a $R$-module free, for all $r \in R$ and $m \in M$ such that $r \neq 0$ e $m \neq 0$ we have $rm \neq 0$. Comments: I tried to do the following: Suppose that $rm = ...
0
votes
1answer
21 views

Let $R$ be a UFD and $p,q,r \in R$. $pq=r^3$ and $\gcd(p,q)=1$ then $p,q$ are cubes up to associates.

I'm not too sure how to prove this statement. This seems like a relatively small problem however I can't for the life of me figure out how to start this so I don't really have any working to show. I ...
0
votes
2answers
39 views

Polynom irreducibility over $\mathbb Z_5$

So, I have polynom over $\mathbb Z_5$: $x^8 - x^7 + 2x^6 + x^5 + 2x^4 + 2x^2 +3x +1$ and I have to find his irreducibile factors. How to do that? I can find his roots by replacing $x$ with $[1]_5, ...
0
votes
3answers
60 views

Primitive Root in Quotient Ring

Find a primitive root of $R[x]/\langle x^4+x+2 \rangle$ where $R$ is the integers mod $3$. Is there a good general stratagy to this sort of thing?
2
votes
1answer
35 views

$\mathbb Q$ Field extension

Consider the Field $F = \mathbb Q(2^{\frac 1 3})$, Is $\sqrt 2 \in F$? I'm trying to figure out how to determine that and similar questions, can you give me a hint or some guidance on how to do that? ...
2
votes
1answer
35 views

If R and S are artinian and finite dimensional algebras respectively, then the tensor product of them is artinian.

Let $R$ be an artinian algebra and $S$ be a finite dimensional algebra over the field $k$. How can i show that $R\otimes_kS$ is artinian? I know that $S$ is also artinian since it is finite ...
1
vote
2answers
73 views

Prove that $R$ has no zero divisors and $R$ has an identity.

Let $R$ be a ring with more than one element such that for each nonzero $a \in R$ there is a unique $b \in R$ such that $aba=a$. Prove that $R$ has no zero divisors and $R$ has an identity.
1
vote
1answer
46 views

Ideals on Rings?How do i Define them?

How do I define all the possible ideals of a given Ring-Set? Example on $Z(m)$. Do I stop when I find enough ideals that their union give's me my given set??
2
votes
3answers
54 views

Polynomials in $\Bbb Q[x]$ with same real root dont have common divisor with degree more than 1

Let $f,g\in \Bbb Q[x]$ polynomyals with the same real root $\alpha \in \Bbb R$. I'm asked wether or not $f$ and $g$ must have a common divisor $h\in \Bbb Q[x]$ with $\deg(h) \geq 1$. I believe that ...
0
votes
3answers
96 views

Quotient rings of polynomial rings

I have come across a quite difficult question while I am studying for a test: Let $F=\Bbb Z[x]/(7,x^2-3)$. Let $u$ denote the image of $x$ under the canonical epimorphism from $\Bbb Z[x]$ to ...
0
votes
2answers
34 views

Greatest common divisor of polynomials over $\mathbb{Q}$

I have two polynomials: $f: x^3 + 2x^2 - 2x -1$ and $g: x^3 - 4x^2 + x + 2$. I have to do two things: find $gcd(f,g)$ and find polynomials $a,b$ such as: $gcd(f,g) = a \cdot f + b \cdot v$. I have ...
2
votes
1answer
33 views

Problem related to Cyclotomic Polynomials

I'm trying to show that if $p$ is prime, then $$x^{p-1}-x^{p-2}+x^{p-3}-...-x+1$$ is irreducible over $\mathbb{Q}$. I don't have an idea of how to start. I know the $p^{th}$ cyclotomic polynomial is ...
2
votes
2answers
38 views

Idempotent Elements of a Commutative Ring

I have to prove this statement and I'm a bit unsure how to go about it: Show that the set of all idempotent elements of a commutative ring is closed under multiplication. Furthermore, find all the ...
1
vote
0answers
54 views

Resolution of module over polynomial ring

The problem is: Let $F$ be a field, and let $R = F[x_1, \ldots, x_r]$, the polynomial ring over $F$. Consider the $R$-module $M = R/(x_1, \ldots, x_r) \cong F$. Find a resolution of $M$ by free ...
0
votes
1answer
35 views

Principal Ideal Domains

I'm trying to teach myself by doing questions. I understand the definition of a ideal is a multiplicatively closed additive subgroup of a ring. And a principal ideal means it has a generator 'g'. So ...
2
votes
2answers
55 views

Example of Localization and Prime Ideals

For each $n\in \mathbb Z^+$, give an example of a localization of $\mathbb Z$ with exactly $n$ prime ideals. Justify your answer. Could an example have something to do with a UFD or Noetherian ...
4
votes
1answer
33 views

Finding the conjugates in $\mathbb{C}$ of a given number over a given field…

I'm having somewhat of a difficult time understand what's being asked—and thus having a hard time answering the question: Find all conjugates in $\mathbb{C}$ of the given number over the given ...
1
vote
2answers
22 views

Size of an ideal in a polynomial Ring

Let $F$ be a field and let $I = \{f(x) \in F[x]\mid f(a) = 0 ~~ \forall a \in F\}$. Prove that $I = \{0\}$ when $F$ is infinite. I have already shown that $I$ is an ideal and that $I$ is infinite ...
0
votes
2answers
30 views

Set or Ring, and group of units?

I have a couple of questions. I understand the axioms needed for a ring. But am confused about a unitary ring? does this just mean its a ring but has to have the unit 1? Also I do not understand ...
2
votes
1answer
79 views

Is my proof for this completion of a ring not being flat correct?

I wanted to show that for $A = K[X_i, i \in \mathbb{N}]/(X_iX_j)_{i,j \in \mathbb{N}}$ the completion $A[T] \rightarrow A[|T|]$ is not flat. However, my proof seems a bit simple/ direct to me, so I'm ...
3
votes
2answers
58 views

Relationship between the characteristic of a ring and a quotient ring

Let $A$ be a ring with unity such that $\operatorname{char} A=8$. Let $I$ be an ideal of $A$. Show that $\operatorname{char}(A/I) \neq 0$ and $\operatorname{char}(A/I)\leq 8$. What I think I know: ...
0
votes
0answers
47 views

How can $\{ a+b \sqrt d : a,b \in \mathbb Z \}$ be a subset of $\mathbb C$?

I got this question on one of my homeworks. However as far as I am aware, $R \subseteq \mathbb C$ only when $ d= -1$. Am I overlooking something that makes it possible in the general case, or can ...
1
vote
3answers
31 views

Show that if $m\in M_n$ and $k \in \Bbb Z_n$ then $mk\in M_n$.

Let $M_n = \{a\in \Bbb Z_n\mid \text{ there exists a non-zero integer $k$ with the property that }a^k\equiv 0 \pmod n \}.$ Show that if $m \in M_n$ and $k \in \Bbb Z_n$, then $mk\in M_n$. For which ...
0
votes
1answer
29 views

Integral multiplicative system over a domain

Suppose $A$ is a domain and $S\subseteq A$ is a multiplicative system. Show that $S\subseteq A^\times$ if and only if $S^{-1}A$ is integral over $A$. I've started $\Leftarrow$ below... Suppose ...
0
votes
3answers
52 views

Maximal Ideal of $\mathbb{Z}[i]$

I'm trying to show that $<1-i>$ is a maximal ideal of $\mathbb{Z}[i]$. I started by assuming there is some ideal $A$ that properly contains $<1-i>$, and then I want to show that $1 \in ...
0
votes
2answers
58 views

Localization of a Dedekind domain.

I have a question on localizations of Dedekind rings which I am learning about in an undergraduate class. Let $R$ be a Dedekind ring with quotient field $K$, $\mathfrak p$ a nonzero prime ideal in ...
3
votes
1answer
128 views

Exercise 17.2 of Matsumura about CM rings

I have two questions about this exercise of Matsumura: Question 1: Why $y^3$ is $R/(x^3)$ regular? Question 2: I hardly (in 20 lines) can prove that $k[x^4,x^3y,xy^3,y^4]$ is not CM. Is there a ...
2
votes
3answers
111 views

Nilpotent elements in $\mathbb{Z}_n$

I'm trying to show that $\mathbb{Z}_n$ has a nonzero nilpotent element if and only if $n$ is divisible by the square of some prime. I have figured out the proof of showing that if $n$ is divisible by ...
2
votes
2answers
58 views

“Primeness” of C[x] in B[x], where A is a subring of B and C is the integral closure of A in B.

Let A be a subring of B, and C the integral closure of A in B. If f, g are monic polynomials in B[x] such that fg is in C[x], then f, g are in C[x]. The first part of the problem allowed the ...
2
votes
1answer
33 views

Units in a polynomial ring

I'm trying to determine $U(\mathbb{R}[x])$, where $U(R)$ denotes the units group of a ring R. I think the answer is all non-zero constant polynomials, but I'm having trouble showing that these are ...
1
vote
0answers
55 views

Class Group of Ring of Integers of $\mathbb{Q}[\sqrt{-57}]$

Let $R$ denote the ring of integers of the imaginary quadratic number field $\mathbb{Q}[\sqrt{-57}]$. I must find the ideal class group $\mathcal{C}$. Using the Minkowski Bound, I know that I need ...
1
vote
1answer
32 views

Homework: If a is a unit, prove that b divides c if and only if ab divides c.

As it says in the title, this is a homework question, so try not to give everything away. I'm just looking for a starting point. The question is stated as follows: Let $R$ be a commutative, unital ...
1
vote
1answer
55 views

Any ring is integral over the subring of invariants under a finite group action

I need to prove that if $G$ is a finite group that acts on ring $A$, and $A^G$ is the subring consisting of elements of $A$ which are invariant under all $g\in G$, then $A$ is integral over $A^G$. ...
0
votes
0answers
17 views

prove that if A is a finitely generated faithfully right $\Bbb R$-module

prove that if A is a finitely generated faithfully right $\Bbb R$-module ,then there exists $\Bbb { B\le A_R}$ such that $\Bbb {A/B} $ is faithful but $\Bbb {A/C}$ is unfaithful for all $\Bbb { ...
2
votes
1answer
46 views

If ring $B$ is integral over $A$, then an element of $A$ which is a unit in $B$ is also a unit in $A$.

Let $A$ be a subring of ring $B$, with $B$ integral over $A$. If $x$ in $A$ is a unit in $B$, then it is a unit in $A$. I know that $f(t) = t - x$ is in $A[t]$ with $f(x) = 0$, and that there ...
3
votes
2answers
87 views

Finite field with algebraic element over $\mathbb{Z}_p$

Let $F$ be a finite field of characteristic $p$. Show that every element of $F$ is algebraic over $\mathbb{Z}_p$. Since char$(F)$ = $p$, $\forall a \in F $ not zero, $ap$ = 0. We also have the same ...
2
votes
2answers
139 views

Field extension of a finite field

Let $E$ be an extension field of a finite field $F$ , where $F$ has $q$ elements. Let $a \in E$ be algebraic over $F$ of degree $n$. Prove that $F(a)$ has $q^n$ elements. I am not sure how to do this ...
3
votes
2answers
168 views

Field extensions and algebraic/transcendental elements

Let $E$ be an extension of field $F$, and let $\alpha, \beta \in E$. Suppose $\alpha$ is transcendental over $F$ but algebraic over $F(\beta)$. Show that $\beta$ is algebraic over $F(\alpha)$. ...
2
votes
1answer
58 views

Show that a set is a ring

Let $R\not=\{0\}$ be a commutative ring with unity. Let $I$ be a prime ideal in $R$. Let $S=R-I=\{x\in R|x\not\in I\}$. Let $F$ be a field that contains $R$ as a subring with the same unity. Show that ...
1
vote
1answer
80 views

Quotient Ring of finite order with root of irreducible polynomial

The question is as follows: Suppose that $\alpha\in\mathbb{C}$ is a root of the irreducible polynomial $f(t) = t^d + \sum_{i=0}^{d-1}a_it^i$ , where $a_i\in\mathbb{Z}$ ($0\le i\le d-1$). Let ...
1
vote
2answers
29 views

Proving C is a Subring of R

For the three axioms Is $0$ contained in C? I got that by putting $a=0$ $(0)(r)=(r)(0) = 0$ For is $a-b$ contained in $C$ and Is $(a)(b)$ contained in $C$ I' ve been playing around with the ...
0
votes
1answer
22 views

If $R$ is $\text{UFD},$ then $R[X,Y]$ is $\text{UFD}.$

Let $R$ be commutative ring with $1.$ Suppose $R$ is $\text{UFD}.$ Could anyone advise me on how to prove $R[X,Y]$ is $\text{UFD}\ ?$ Thank you.
2
votes
0answers
33 views

some ring theory applied to holomorphic functions

I'd like to know if my understanding of this business is correct. Let $U \subset \mathbb{C}^n$ be open and connected. The set $\mathcal{K}(U)$ of meromor­phic functions on $U$ is a field. Is it true ...
0
votes
0answers
42 views

Why is this not trivial? - $f(x)~|~g(x) \iff g(x) \in \langle f(x) \rangle$

Let $F$ be a field and $f(x), g(x) \in F[x]$. Show that $f(x)$ divides $g(x)$ if and only if $g(x) \in \langle f(x) \rangle$. This seems... almost trivial to me (which is usually a sign that I'm ...
1
vote
1answer
89 views

Polynomial rings over a field and maximal/prime ideals

Let $F$ be a field , I want to prove that every proper nontrivial prime ideal of $F[x]$ is maximal. My definitions of prime/maximal ideals are as follows: $N$ is a prime ideal of $R$ iff $ab \in N ...