1
vote
3answers
27 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
24 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
44 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
38 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 ...
2
votes
3answers
57 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
54 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
30 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
27 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
23 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
0answers
33 views

Ring A 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 $A^G$. The hint ...
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
39 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
75 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
78 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
94 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
50 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
56 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
27 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
20 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
26 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
41 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
51 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 ...
1
vote
3answers
68 views

Finite rings and subrings isomorphic to $\mathbb{Z}_n$

My book has proven this: Every ring with unity has a subring isomorphic to either $\mathbb{Z}$ or $\mathbb{Z_n}$. The $\mathbb{Z_n}$ case arises if the parent ring has characteristic $r>0$ I ...
1
vote
1answer
42 views

Simple $R$-module

Let $M$ be a simple $R$-module and $N=M\bigoplus M$. Then which one is true: 1) $N$ has a finite number of submodules. 2) $\operatorname{Hom}_R(N,N)$ is a division ring. 3) ...
0
votes
1answer
41 views

Maximal and prime ideals of $\mathbb{Z} \times \mathbb{Z}$

I have to find a maximal ideal of $\mathbb{Z} \times \mathbb{Z}$ , and a prime ideal that is NOT maximal. Or, essentially, I want $I$ such that $\mathbb{Z} \times \mathbb{Z} / I$ is a field, and I ...
0
votes
0answers
22 views

Primitives in unique factorisation domains

Problem Let R be a UFD, $f = a_0 + a_1 X + ... + a_n X^n \in R[X]$, where f is primitive. Suppose $p \in R$ is irreducible with $p \mid a_i$, $0 \le i \lt n$, $p^2 \nmid a_0$, $p \nmid a_n$. To show ...
1
vote
0answers
34 views

prove that the ideal $I=(x,xyx,\dotsc,xy^nx,\dotsc)$ cannot be generated by a finite number of polynimials.

The ring is $K\langle x,y \rangle$ So we have to assume that exist a finite number. Let $f_1,\dotsc,f_m$ a finite number of polynomials. How we can arrive at a contradiction?
0
votes
1answer
173 views

Homogeneous and maximal ideal in a $\mathbb Z$-graded ring

Is Exercise 2.8 from Marley's notes on "GRADED RINGS AND MODULES" true? Exercise 2.8: Let $R$ be a graded ring and $M$ a homogeneous maximal ideal of $R$. Prove that $M =…⊕R_{-1}⨁m_0⨁R_1⨁…$, ...
2
votes
1answer
49 views

If $S$ be a ring with identity and $f: M_{2}(\mathbb{R})\longrightarrow S$ is a nonzero ring homomorphism, which cases is true?

Let $S$ be a ring with identity and $f: M_{2}(\mathbb{R})\longrightarrow S$ is a nonzero ring homomorphism,then which cases is true? $S$ is left Artinian $S$ is left Noetherian $S$ is simple ring ...
3
votes
1answer
80 views

Describe all ring homomorphisms from Z x Z to Z x Z

Note: In this class, a ring homomorphism must map multiplicative and additive identities to multiplicative and additive identities. This is different from our textbook's requirement, and often means ...
1
vote
1answer
73 views

Ring homomorphism and ideals

Let $R$ and $R'$ be rings and let $\phi: R\mapsto R'$ be a ring homomorphism and $N$ an ideal of $R$. Show that $\phi[N]$ is an ideal of $\phi[R]$, and give an example to show that $\phi[N]$ need not ...
1
vote
3answers
57 views

R is a commutative ring with unity and prime characteristic p, show that $\phi: R \to R\,\,/\,\, \phi(a) = a^p$ is a homomorphism

It's pretty obvious that $\phi(0) = 0$ and $\phi(1) = 1$ so those are all set. Now I want to show that $\phi(a+b) = \phi(a) + \phi(b)$ or $(a+b)^p = a^p + b^p$ for all $a,b \in R$. however it's ...
2
votes
2answers
49 views

An integral domain $A$ which is also absolutely flat is a field

Question: Assume that $A$ is an integral domain such that every $A$-module is flat. Show that that $A$ is a field. Discussion: This seems to be very related to this question, in which it is shown ...
5
votes
2answers
85 views

When is a local, reduced, (commutative) ring an integral domain?

Question I am wondering whether or not it is true that if $A$ is a reduced ring, then is it the case that the localization of $A$ at any of its prime ideals is an integral domain? Discussion ...
2
votes
5answers
161 views

Maximal ideal contains a zero divisor

Suppose $R$ is a commutative and unital ring. Let the ideal $I$ be maximal and $a,b$ be (nonzero) zero divisors in $R$. Show that $ab = 0$ implies $a \in I$ or $b\in I$ We've ...
0
votes
2answers
44 views

Let $f(x)$ belong to $\mathbb{Z}_p[x]$. Prove that if $f(b)=0$, then $f(b^p)=0$. [duplicate]

Let $f(x)$ belong to $\mathbb{Z}_p[x]$. Prove that if $f(b)=0$, then $f(b^p)=0$. Not sure how to proceed with this problem. I usually use Chegg, but Chegg doesn't have the solution for this problem. ...
0
votes
2answers
28 views

Common divisors in a PID

Suppose that R is an integral domain and that α, β, γ ∈ R. We say that γ is a common divisor of α and β if γ|α in R and γ|β in R. Suppose that R is a PID. Suppose that α, β ∈ R. Let I = (α) and ...
3
votes
2answers
81 views

Orthogonal idempotents from disjoint union in $\text{Spec}(A)$

Let $A$ be a commutative ring with unity. Suppose that $X=\text{Spec}(A)$ is a disjoint union $X_1\cup X_2$ of topological spaces. Show that $A$ has a pair of orthogonal idempotents $e_1,e_2$ ...
1
vote
2answers
140 views

Prove: The pre-image of an ideal is an ideal.

Let $\phi : R \to S$ be a homomorphism. If $N$ is an ideal of $S$, then $\phi ^{-1} (N)$ is an ideal of $R$.
2
votes
1answer
50 views

Nilpotent elements in a commutative ring

Let $R$ be a commutative ring. Show that for any $a,b \in R$ nilpotent that $a+b$ is also nilpotent in $R$. We know $a^n = 0$ for some n and $b^m = 0 $ for some m, so consider $(a+b)^{m+n} = ...
1
vote
1answer
40 views

Principal prime ideal is generated by irreducible element

$R$ is an integral domain, $x\in R$ and $(x)=I$ is a prime ideal. Prove that $x$ is an irreducible element of $R$. So I assume $ab\in I$, with $a, b \in R$. Since $I$ is a prime ideal, either $a$ ...
1
vote
1answer
41 views

Kernel and image of map on unit group of finite field

This question is from a rings and modules course I'm doing: Let $p>2$ be prime and $f:\mathbb{F}_{p}^{*}\rightarrow \mathbb{F}_{p}^{*}$ be given by $x \mapsto x^\frac{p-1}{2}$. Prove that $ker(f)$ ...
2
votes
1answer
41 views

idempotent functions in the space of continuous functions

question :$X$ is a connected iff the zero and identity functions are the only idempotent function in $C(X)$ answer: if $f\in C(X)$ is idempotent then $f(x)^2=f(x)$ $\forall x\in X $ then $f(x) =0 ...
4
votes
1answer
52 views

Ideal contained in the union of two ideals and a prime

Taken from Miles Reid "Undergraduate Commutative Algebra" p.35 ex. 1.12 b) Let $I,J_1,J_2 \subset A$ be ideals of a commutative ring $A$. Let $P$ be a prime ideal, then if $I \subset J_1 \cup J_2 ...
2
votes
0answers
41 views

If there is an integer $n$ such that $n^2\equiv3\pmod p$, where $p$ is prime, prove there are integers $a$ and $b$ such that $|a^2-3b^2|=p$

If there is an integer $n$ such that $n^2\equiv3\pmod p$, where $p$ is prime, prove there are integers $a$ and $b$ such that $|a^2-3b^2|=p$. So $n^2-3 = pm$ for some integer $m$, and I know $|a^2 ...
5
votes
1answer
62 views

Find a maximal ideal $I$ in the ring $\mathbb{Z}[i]$ such that $\mathbb{Z}[i]/I$ is isomorphic to $\mathbb{Z}/521\mathbb{Z}$.

I know $\mathbb{Z}[i]$, the Gaussian integers, is a PID. So $I$ is generated by a single element. At first I thought $I=(521)$, but $521$ can be reduced to $11^2 + 20^2$. Would $I=(11 + 20i)$ or ...
0
votes
1answer
26 views

Ideal of a field

Let $F$ be a field. Show that $S$ be a non empty subset of $F^{n} $ then $ I(S) =$ { $ f(x) \in F[x] \hspace{0.1in} \vert \hspace{0.1in}f(s) = 0 \hspace{0.1in} \forall s \in S $ } is an ...
2
votes
1answer
96 views

Triangular Matrices and Simple Modules

Let $\Bbb{T}_n(k)=\{n \times n \text{ upper triangular matrices (which includes the diagonal entries)}\}$ I want to first express $\Bbb{T}_{n}(k)$ (as a $\Bbb{T}(k)$-module) as a direct sum ...
0
votes
2answers
60 views

Number of elements in Quotient ring over $\mathbb Z_5[x]$ and $\mathbb Z_{11}[x]$

a) How many elements does the quotient ring$\displaystyle \frac{\mathbb Z_5[x]}{\langle x^2+1\rangle} $ have? Is it an integral domain? I can see that the polynomial, $\displaystyle p(x)= ...
1
vote
1answer
54 views

Finish a proof that every prime ideal of a ring is the contraction of a prime ideal in its formal power series

Given a commutative ring $A$ with identity, and its formal power series ring $A[[x]]$, I am attempting to prove that every prime ideal of $A$ is the contraction of a prime ideal of $A[[x]]$. ...