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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$A$ 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
95 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.
2
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1answer
76 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 ...
1
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0answers
145 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. ...
5
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3answers
136 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
26 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
59 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
46 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 ...
2
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2answers
61 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 ...
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1answer
46 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
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2answers
509 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 ...
0
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2answers
36 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
75 views

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
76 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
99 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 ...
2
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1answer
26 views

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
67 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
139 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
47 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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1answer
57 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
59 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 ...
6
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3answers
313 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
54 views

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? ...
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1answer
34 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
votes
1answer
150 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 & ...
3
votes
2answers
244 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 ...
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0answers
159 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
44 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
49 views

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 ...
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1answer
57 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
49 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 ...
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2answers
170 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 ...
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2answers
62 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
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1answer
34 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 ...
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1answer
51 views

Verification of Proof: Let $R$ be a ring of unity and $a \in R$ satisfy $a^2=1$. $S=\{ara \mid r \in R\}$ is a subring

Here's what I got. The three conditions we have to prove are: $0$ is in $S$: Let $r=0$ and this implies $a0a=0a=0$ which is in $S$ $(a-b)$ is in $S$ for all $a,b \in S$: Let $a, b \in S$. this ...
1
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2answers
25 views

$x$ in intersection of maximal ideals implies $1-x$ is a unit

Let $R$ be a commutative ring, we define $J:=\bigcap_{\mathcal M \space \text{maximal}}\mathcal M$. Let $x \in J$, prove the following $(1-x) \in \mathcal U(R)$ If $x^2=x$ then $x=0$ For the ...
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2answers
50 views

Computing the center of a ring

if i have the following ring $R = \mathbb{H} \otimes _\mathbb{R} M_2(\mathbb{C}) $ then how would i find the center $Z(R)$? Also is this ring simple, i am sure it is but am struggling to show that ...
2
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3answers
91 views

inverses in $R/I$ where $I$ is a nilpotent ideal

Given an element $x \in R$ where R is a ring $I$ is a nilpotent ideal of $R$, i am trying to find inverses in the quotient R/I and thought about things in the general case, what would determine the ...
2
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2answers
73 views

A question about von Neumann regular rings and their ideals

Suppose $R$ is a von Neumann regular commutative ring with a unit. Prove that every principal ideal $I$ is generated by an idempotent element and for every principal ideal $I$, there exists a ...
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1answer
117 views

Discrete Valuation Rings - Atiyah & MacDonald

The following is claimed (without much proof) during the the proof of Prop 9.2 in Atiyah & MacDonald. Saurabh commented below giving the proof that was probably intended by A&M (thank you!). I ...
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1answer
227 views

When is the quotient ring of a multivariable polynomial ring over a field by a monomial ideal an integral domain?

When is the quotient ring of a multivariable polynomial ring over a field by a monomial ideal an integral domain? I am actually trying to show that a monomial ideal is prime by showing the ...
0
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1answer
85 views

Questions concerning $\mathbb Z_3[x]/(x^3+2x-1)$

Is the automorphism group of $\mathbb Z_3[x]/(x^3+2x-1)$ cyclic ? Is $\mathbb Z_3[x]/(x^3+2x-1)$ separable ?
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1answer
41 views

Should Ext-quiver be a full sub-quiver of its AR-quiver for a basic hereditary algebra A over algebraic closed field K?

For a basic hereditary algebra A over algebraic closed field K, prove its Ext-quiver $\Gamma_{A}$ is a full sub-quiver of its AR-quiver $\Delta_{A}$. I have no clue for this.
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0answers
86 views

Henselization of the ring of polynomials

I am trying to understand example of Henselization from wiki. http://en.wikipedia.org/wiki/Henselian_ring#Henselization It says that Henselization of the ring of polynomials localized at point $(0, ...
3
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2answers
118 views

Prove every maximal ideal of $A$ is a prime ideal (Hint: Use the fact that $J$ is a maximal ideal iff $A/J$ is a field.)

Let $A$ be a commutative ring with unity. Prove: Proof every maximal ideal of $A$ is a prime ideal (Hint: Use the fact that $J$ is a maximal ideal iff $A/J$ is a field.) In the question before ...
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1answer
41 views

Factor $55 - 88 \sqrt{-2}$ as a product of primes in $\mathbb{Z}[\sqrt{-2}]$

To solve this problem, I let $K = \mathbb{Q}(\sqrt{-2})$, and I thought to take the norm $$N(55 - 88 \sqrt{-2}) = 55^2 + 2 \cdot 88^2 = 18513 = 3^2\cdot11^2 \cdot 17$$ If $a \in \mathbb{Z}[\sqrt{-2}]$ ...
4
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2answers
184 views

In a Noetherian integral domain, a principal prime ideal can't have proper non-zero prime ideals

Let $R$ be an integral domain and Noetherian. Let $P \subset R$ be a non zero prime ideal. Prove that if $P$ is principal then there is no prime ideal $Q$ such that $0 \subsetneq Q \subsetneq P$. ...
0
votes
1answer
98 views

$\mathbb R[X]/\langle X^4-1\rangle \cong \mathbb R \times \mathbb R \times \mathbb C$

I am trying to prove the isomorphism $\mathbb R[X]/\langle X^4-1\rangle \cong \mathbb R \times \mathbb R \times \mathbb C$. I will write what I did so you can help me from there. First notice that ...
1
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3answers
154 views

Calculate the dimension of the field extension $[\mathbb{Q}[ \sqrt2 , \sqrt3] : \mathbb{Q}]$

I've though that $[\mathbb{Q}[ \sqrt2 , \sqrt3] : \mathbb{Q}] = [\mathbb{Q}[ \sqrt2] : \mathbb{Q}].[ \mathbb{Q}[\sqrt2, \sqrt3]:\mathbb{Q}[ \sqrt2] ] $ And I know how to prove $[\mathbb{Q}[ \sqrt2] : ...
2
votes
1answer
70 views

Begginer doubt in Ring of p-adic integers

I am studying $p$-adic Rings and let me explain my understanding and doubt here. As I understood, Let $p$ be a rational prime and $Z$ denotes ring of integers, then form cartesian product $$P=Z/pZ ...