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
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Module and Noetherian/Artinian Rings

I am trying to prove that: Every finitely generated $F$-module $M$ is both Noetherian and Artinian where $F$ is a field. For this I am looking at the submodules of $F$ and saying that they are in ...
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2answers
20 views

Let R be an integral domain. Show that if the only ideals in R are {0} and R itself, R must be a field [duplicate]

I know that if (x)={0} then the if 0=r0 such that r belongs to R therefor it's a field. Most likely I'm wrong but I need help with the second part if the ideal is R
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1answer
24 views

Intersection of localizations of an integral domain

I have a few questions about proving the following identities: $$\bigcap_{p \in SpecA}A_p = A \ \ \ \ \bigcup_{p \in SpecA}A_p = K$$ Here $A$ is an integral domain, $K$ is its field of fractions. ...
2
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2answers
38 views

A non-projective module

Let a ring with identity $R$ be decomposed as $S_1⊕\cdots⊕S_n$ (as a right $R$-module), where $S_i=e_iR$ with $e_i$ nonzero idempotents of $R$ adding up to $1$. If $J$ is the Jacobson radical of ...
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1answer
52 views

Confusion about the definition of localization

For example, let $\mathbb Z$ be the ring and $S = \mathbb Z - 2\mathbb Z$. Then the quotient ring should be: $S^{-1}A = \{a/s: a\in \mathbb Z \text{ and }s \in S\}$, which is formed by the equivalence ...
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2answers
49 views

Proving Fermat's Little Theorem in general and use that to prove Euler's Generalization of Fermat's Little Theorem

Can anyone help me with this? I know there are many different ways to do this and threads explaining this question. However I can't seem to find one that uses only group/ring theory. I haven't ...
2
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1answer
39 views

Equalities in the categories of modules

It is well-known that over a quasi-Frobenius ring $R$ any f.g. right module $M$ is reflexive, in the sense that whenever we take $M^*=Hom_R(M,R_R)$ as a left $R$-module, the modules $M$ and $M^{**}$ ...
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0answers
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Proof of Version of Krasner's Lemma

I need some help in constructing the proof to this version of Krasner's Lemma from Serre's Local fields text book: Let E/K be a finite Galois extension of a complete field K. Prolong the valuation ...
2
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0answers
22 views

Polynomials in the extension of a complete field

I need some help in answering this exercise from Serre's Local Fields textbook: Let K be a complete field, and let f(X) in K[X] be a separable irreducible polynomial of degree n. Let L/K be the ...
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2answers
59 views

Counterexample for non-prime ideal [closed]

Let $A$ be a commutative ring and $P$ be a prime ideal. Then $S = A - P$ is a multiplicatively closed subset. My question is if it is true that $S$ is a multiplicatively closed subset, then $A-S$ ...
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0answers
47 views

Prime ideals in $A$ and prime ideals in $S^{-1}A$

Let $A$ be a ring and $S$ be a multiplicative closed subset. Then there is a 1 to 1 correspondence between the prime ideals in $A$ (intersect $S$ is empty) and prime ideals in $S^{-1}A$. My question ...
2
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3answers
52 views

$k[t^{a_1},t^{a_2},t^{a_3}]$ in the form $k[x,y,z]/(…)…(…)$

I want to write $k[t^6,t^7,t^{15}]$ in the form $k[x,y,z]/(...)...(...)$; but I even don't know how to start. is there in general a way that one can write $k[t^{a_1},t^{a_2},t^{a_3}]$ in the ...
-1
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2answers
34 views

Find kernel of homomorphism of rings

$\phi: \mathbb Q[x] \to Mat_{2,2}(\mathbb Q): \phi(a) = \begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix}, \phi(x)= \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}$. Find kernel of $\phi$, ...
-1
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1answer
47 views

Does there exist a non-commutative ring of order $210$? [closed]

Does there exist a non-commutative ring of order $210$? One can construct rings of this order by taking products of $\mathbb Z/n\mathbb Z$, but those are commutative. Historical aside (not ...
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2answers
40 views

$A$ is a commutative ring, $P$ is a prime ideal. Prove $A_P$ is local ring

Question: Suppose $A$ is a commutative ring, $P$ is a prime ideal. Prove $A_P$ is local ring. I have no idea how to construct the unique maximal ideal.
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3answers
37 views

Show that $\mathbb{Z}_n$ is local ring iff $n$ is a power of a prime number [closed]

$\mathbb{Z}_n$ is integers modulo $n$. Local ring is a commutative ring if it has a unique maximal ideal. Please help me prove the claim.
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0answers
60 views

Ideals, prime ideals and maximal ideals of the ring $K=\mathbb R[x]/\langle (x^2+1)(x-2)^2\rangle$ [closed]

I am trying to find the ideals, prime ideals and maximal ideals of this ring: $K=\mathbb R[x]/\langle (x^2+1)(x-2)^2\rangle$. I am fairly fluent in abstract algebra though ideals are my huge ...
2
votes
1answer
50 views

Characteristic of a Ring not making sense.

The characteristic of a ring with unity is defined to be the least positive integer $n$ such that $1$ plus itself $n$ times $=0$. How does this make sense? $1$ plus itself $n$ times $=n1=n=0$, but ...
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0answers
70 views

Ambiguity in the definition of unmixed ideal

Compare the definitions: Page 136 Matsumura, Commutative ring theory: A proper ideal $I$ in a Noetherian ring $A$ is said to be unmixed if the heights of its prime divisors are all equal. ...
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3answers
89 views

A good introductory book on Ring and Field theory with a view towards Number Theory ?

Please suggest some good introductory books on Rings&Fields with a view towards Number Theory ?
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1answer
25 views

1 is gcd(x,y) but 1 cannot be expressed as ax+by where a,b,x,y are in Z[sqrt(-5)].

I am working in the ring Z[sqrt(-5)]. I have shown that 1 is a gcd(x, y) where x=3 and y=2+sqrt(-5). I would like to show however that 1 cannot be expressed as ax+by where a, b are in Z[sqrt(-5)]. I ...
2
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1answer
46 views

What happens with $S_n$ in rings, integral domains and fields?

From Cayley's theorem we know that every group is a symmetric group, i.e. a group of permutations. But what happens when we "extend" a group to a ring or a field for example; is there any ...
2
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1answer
31 views

Is an ideal prime when its complex extension is prime?

Let $I = \langle f_1,\dots,f_k\rangle$ be an ideal in $\mathbb R[x_1,\dots,x_n]$. The same $f_i$ generate an ideal $\widetilde I$ in $\mathbb C[x_1,\dots,x_n]$. When $\widetilde I$ is prime in ...
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2answers
50 views

Behavior of the annihilators of modules via monic and epic homomorphisms

Let $f:M\rightarrow N$ be an $R-$ homomorphism. Prove that if $f$ is monic, then $l_{R}\left(M\right)\supseteq l_{R}(N)$ , whereas if $f$ is epic, then $l_{R}(M)\subseteq l_{R}(N)$ . This is ...
0
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1answer
37 views

Assume G is a group, x,y is in G; x and y are not identity, but $x^3=1$ and $y^2=1$ and $(xy)^2=1$. Find the order of G and the group table

So I am stuck with this problem and I can't seem to find the relationship with the x, y and identity in dealing with size of group and how they connect with $(xy)^2=1$. Can someone help me with this? ...
2
votes
2answers
27 views

Prove that $I \subseteq R$ is prime if and only if $R/I$ is an integral domain.

We say that an ideal $I \subseteq R$ is prime if for all $a, b \in R$, $ab \in I$ implies that $a \in I$ or $b \in I$. (a) Prove that $I \subseteq R$ is prime if and only if $R/I$ is an integral ...
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2answers
30 views

Prove that $U = t · \mathbb{R}[t]$ is a maximal ideal in $\mathbb{R}[t]$

I was studying for an exam and chanced upon this question in my textbook. I was a bit confused as to how we would go about trying to solve it. Any help would be appreciated! :) Prove that $U = t · ...
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2answers
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No polynomial of degree 3 in $\mathbb{R}[x]$ is a prime

How do I prove that no polynomial of degree 3 in the ring $\mathbb{R}[x]$ of polynomials with real coefficients is a prime? I really need help on this one guys. I know I need to use the Intermediate ...
1
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1answer
30 views

Kernel of formal differentiation in a field of characteristic $0$

From the previous parts I've proven that $$D:F[x]\to F[x]$$ is an additive group homomorphism on addition for $F[x]$ and not a ring homomorphism because the multiplication does not hold. OK, so I ...
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1answer
44 views

Show that $Rad(I)$ is a prime ideal

The ring $R$ is commutative with unit. An ideal $I$ is called primary, if it stands the following: If $ab \in I$ then $a \in I$ or $b^n \in I$, for a natural number $n$. Show that if $I$ is a ...
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1answer
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If z is a common divisor of x and y then z is a unit.

In the ring $Z[\sqrt{-5}]$, I would like to show that if $z$ is a common divisor of $x=3$ and $y=2+\sqrt{-5}$, then $z$ is a unit. I know that I will have to prove that $N(z)=1$, am I right in ...
2
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2answers
44 views

A divisor of a unit is a unit?

Is it true that if $ab=u$ where $u\in U(R)$ is a unit of the noncommutative ring $R$, then $a,b\in U(R)$? If $R$ is commutative, then this can be seen by $$a(bu^{-1})=uu^{-1}=1=(bu^{-1})a,$$ but ...
4
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1answer
66 views

Does any (noetherian) integral domain have a “UFD closure”?

Let $R$ be a (possibly noetherian if that helps) commutative unital integral domain. Does there exist a UFD $\overline{R}$ such that $R$ embeds in $\overline{R}$ (via some map $\psi$) and such that ...
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0answers
27 views

Homomorphic images of $\mathbb{Z}[x]$

How to prove that any finite field is a quotient ring of $\mathbb{Z}[x]$ ? I am not sure whether this result is true or false. Any hint will be appreciated. Thanks in Advance.
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0answers
28 views

number of elements in the quotient ring $\frac{\mathbb{Z}_n[x]}{<ax+b>}$

How to find the number of elements in the quotient ring $\frac{\mathbb{Z}_n[x]}{<ax+b>}$ where n is a composite number. In particular what is the number of elements in the ring ...
1
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1answer
49 views

Example of a semi-simple $\mathbb{R}$ algebra

Let $[n]:=\{1,....,n\}$ and define the $2^n$-dimensional $\mathbb{R}$-algebra $C_n$ with basis $e_I$, $I \subset [n]$, such that $e_\emptyset = 1, e_ie_j = -e_je_i$ for $i \not =j, e_j^2 = 1 $ and ...
0
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1answer
13 views

Explaining a. Q(R,T) has unity even if R does not and b. In Q(R,T) every nonzero element of T is a unit

So I've been having trouble understanding this proof for quite a while now. I understand how the field of quotients is formed but not so much of why my professor's answers use this tactic for this ...
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3answers
94 views

Show that quotient rings are not isomorphic

I've been given a homework problem that requires me to show that the rings $\mathbb{C}[x,y]/(y - x^2)$ and $\mathbb{C}[x,y]/(xy-1)$ are not isomorphic. This is my attempt at a solution: For ...
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2answers
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Localization Question: $\frac{a}{b}\in\left(\mathbb{Z}_{(p)}\right)^{\times}\iff\frac{b}{a}\in\mathbb{Z}_{(p)}$

Questions: $\rm\color{#c00}{(1)}$ Is the $[\Longrightarrow]$ implication of $$ \frac{a}{b}\in\left(\mathbb{Z}_{(p)}\right)^{\times}\iff\frac{b}{a}\in\mathbb{Z}_{(p)} $$ obvious? ...
1
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1answer
29 views

showing Module is simple

Given the following: let $C \subset \mathbb{H}$ be a subring of the real quarternion algebra such that it contains the center of $\mathbb{H}$ = $Z(\mathbb{H})$ Also C $\cong \mathbb{C}$ Then let R ...
2
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1answer
43 views

Jacobson radical of a certain ring of matrices

Given a Matrix $A \subset M_4(\mathbb{C})$ be the $\mathbb{C}$-subalgebra consisting elements in the form \begin{pmatrix} * & * & * & *\\ * & * & * &*\\ 0 & 0 & ...
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2answers
56 views

Jacobson radicals of $R$ and $R/I$ where $I$ is a nilpotent ideal.

Out of interest If i have the map $\phi: R \longrightarrow R/I $ where $R$ is a ring and $I$ is a nilpotent ideal ? then would i be right in saying that if i were to apply this map to the jacobson ...
2
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0answers
33 views

To find all Ring homomorphisms from $\mathbb Z_m$ to $\mathbb Z_n$ [duplicate]

How to find all Ring homomorphisms from $\mathbb Z_m$ to $\mathbb Z_n$ (with the usual ring structure ) ?
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0answers
39 views

Equivalence of Modules

The category of modules over a ring can be viewed as an enriched version of an action of a monoid on a set (see nLab entry). Moreover, if $R$ is a commutative ring, the category of modules over it is ...
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2answers
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Assume we have $\mathbb{Z}_{p}[x]$ with $p$ being a prime. Prove that $x^{p-1}-1=(x-1)(x-2)…(x-(p-1))$

I know how this formula works and it is quite interesting actually but how would you prove this relationship? Through induction (seems difficult since there's no equation for prime numbers), but I'm ...
0
votes
1answer
26 views

Verification of proof that for a,b in ring R, assuming ab is a zero divisor at least one of a and b is zero divisor

I'm not so sure if this is correct but here's what I have so far: ab is a zero divisor iff there is a c$\neq$0 s.t. (ab)c=c(ab)=0 given ab$\neq$0 and c$\neq$0. Then we have ...
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0answers
45 views

Question on an $\mathbb{R} $-algebra

Define $[n] = \{1,\ldots, n\} $, where $n \in \mathbb{N}$ and define the $2^n$- dimensional $\mathbb{R}$-algebra $C_n$ as follows: Notation: Basis is $e_I$, where $I \subset \mathbb{N}$ and let ...
1
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2answers
39 views

Prove that for $f,g\in F[x]$, where $F$ is an infinite field, if $f(a)=g(a)$ for infinitely many elements $a\in F$, then $f=g$

Prove that for $f,g\in F[x]$, where $F$ is an infinite field, if $f(a)=g(a)$ for infinitely many elements $a\in F$, then $f=g$. I'm not entirely sure how to tackle the "infinitely many elements ...
0
votes
2answers
49 views

Verification of proof that if $R$ is a commutative ring, $a$ is a unit and $b^2=0$ then $a+b$ is also a unit

Here's what I have so far and I would like to know if I am right or if my proof needs to be edited: Since $a$ is a unit it means $a1=a$, with $1$ being the unity element We know $b^2=0$ and this ...
0
votes
1answer
29 views

How many polynomials in $Z_{p}[x]$ have degree n or less?

For your reference, $Z_{p}[x]$ refers to the set of all polynomials with coefficients integer mod p. To me it seems like this and the degree (power) of the two polynomials are unrelated. What ...