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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Suppose a and b belong to a commutative ring and ab is a zero-divisor. Show that either a or b is a zero-divisor.

Suppose a and b belong to a commutative ring and ab is a zero-divisor. Show that either a or b is a zero-divisor. My answer goes like this: If ab is a zero-divisor, then there exists a nonzero ...
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1answer
40 views

Minimal prime ideal

I was trying the following: Let $R$ be a commutative ring with identity, then $R$ has a unique prime ideal if and only if $R$ has a minimal prime ideal which contains all zero-divisors, and all ...
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2answers
33 views

Let $\pi$ be a prime element in $\Bbb{Z}[i]$. Then $N(\pi)=2$ or $N(\pi)=p$ s.t. $p$ is prime and $p\equiv 1\pmod 4$

Let $\pi$ denote a prime element in $\Bbb{Z}[i]$ such that $\pi\not\in \Bbb{Z},i\Bbb{Z}$. Prove that $N(\pi)=2$ or $N(\pi)=p$, where $p$ is a prime number $\equiv 1\pmod4$. Give a complete ...
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1answer
81 views

Prove that $R = S_1$ or $R = S_2$

Let $R$ be a ring and $S_i$ be the subrings of $R$ such that $R = S_1 \cup S_2$. Prove that $R = S_1$ or $R = S_2$. I am not exactly sure what to do here. If I want to proceed with letting $R \neq ...
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2answers
41 views

Two definitions of Jacobson Radical

I have in my notes that the Jacobson radical of a ring $R$ is: $J(R) = \cap${$I$ | $I$ primitive ideal of $R$} $= \cap$ {$Ann_R M$ | $M$ simple $R$-module}. I have now seen elsewhere that J(R) = {x ...
0
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1answer
50 views

For a commutative ring $R$, the set of functions $f:R\rightarrow R$ is a comm. ring

Let $R$ be a commutative ring and let $F=F(R,R)$ be the set of functions $f:R\rightarrow R$. Prove that $F$ is a commutative ring. I want to show that $(F,+)$ is an abelian group. The easy ...
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1answer
21 views

Suppose $R$ is a PID and $ann(M) = (r)$ for some $r \in R$. Prove that for any $x \in R$ if $gcd(x, r) = 1$, then $xM = M$ and $M(x) = \{0\}$.

Let $R$ be a commutative ring and $M$ be an $R$-module $M$. For any $r \in R$, let $rM = \{ rm : m \in M\}$ and $M(r) = \{m \in M: rm = 0\}$. Suppose $R$ is a PID and $ann(M) = (r)$ for some $r \in ...
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1answer
34 views

Associated primes of an $R$-module

An associated prime of an $R$-module $M$ is an ideal of the form $Ann_R(N)$ where $N$ is a prime sub-module of $M$ in the sense that $N$ is nonzero and $Ann_R(N)=Ann_R(N')$ for each nonzero sub-module ...
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1answer
21 views

Automorphism in splitting field

Suppose $F\subseteq L$ is any field extension, $f(x) \in F[x]$ and $\beta_1,\beta_2,....\beta_r\in L$ are distinct roots of $f(x)$. Prove a)If $\sigma$ is an automorphism of L that leaves F fixed ...
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1answer
31 views

Maximal ideals of commutative Artinian rings

I would like some help on an exercise I thought I had done correctly at first glance, but obviously have doubts about. The question is; Let $R$ be a commutative Artinian ring. Then R has finitely ...
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2answers
30 views

Let R be a PID, let (n) be an ideal generated by $n \in R$, how to show that all ideals of R/(n) is of the form (x + (n)) where x divides n? [closed]

Let R be a PID, let (n) be an ideal generated by $n \in R$, how to show that all ideals of R/(n) is of the form (x + (n)) where x divides n? (this is inspired by all ideals of $Z_n$ is of the form (x) ...
2
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2answers
146 views

Why did they need to say that the image is a subset?

I'm reading the course notes on rings at the moment, but noticed something that didn't quite make sense immediately to me. Suppose $R$ is a ring and $f : R \to S$ is a homomorphism. Let $T = ...
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2answers
46 views

Maximal ideals and Prime ideals.

Ok, I am new to the concepts of maximal ideals and prime ideals. I know the definitions for both, but I am kind of stuck with understanding the examples. So, any help would be much appreciated. ...
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3answers
25 views

Z_{p^n} is a local ring

I was trying to prove that: if p is prime and n>1, then Z_{p^n} is a local ring with unique maximal ideal (p). I was trying to show that (p) consists of all nonunits. How to show that the elements ...
3
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2answers
55 views

Every finite ring with identity $a+a = 0$ is subring of $Mat_{n\times n} (\mathbb{F}_2)$ for some $n$?

Is it true, that every finite ring $R$ with identity $\forall a \in R (a+a=0)$ is subring of $Mat_{n \times n}(\mathbb{F}_2)$ for some $n$?
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3answers
89 views

How to show that the ideal $(X^{3},XY,Y^{n})$ of $K[X,Y]$ is primary?

I'm working on a problem in Sharp's Steps in commutative algebra, to be precise exercise 4.28 which states the following: Let $K$ be a field and $R = K[X,Y]$ be the polynomial ring in the ...
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2answers
29 views

Minimal polynomial problem

Show that $\mathbb Q(\sqrt 2 + i)=\mathbb Q(\sqrt 2, i)$ and find minimal polynomial. My question: Assume that they are equal, then the minimal polynomial of both sides must be the same. To prove and ...
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0answers
35 views

Relating $N(α)$ to $|α|$

Consider $\mathbb Q[\sqrt{-1}]$. What would be an equation relating $N(α)$ to $|α|$ which is the natural absolute value defined for complex numbers? For which $\mathbb Q[\sqrt{d}]$ is this formula ...
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0answers
29 views

$\mathbb Q[\sqrt{d}]$ has elements $α$ with negative norm $N(α)$

Assuming that $d$ is not a perfect square, for which integers $d$ the field $\mathbb Q[\sqrt{d}]$ has elements $α$ with negative norm $N(α)$?
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2answers
59 views

Quadratic Integers in $\mathbb Q[\sqrt{-5}]$

Can someone tell me if $\frac{3}{5}$, $2+3\sqrt{-5}$, $\frac{3+8\sqrt{-5}}{2}$, $\frac{3+8\sqrt{-5}}{5}$, $i\sqrt{-5}$ are all quadratic integers in $\mathbb Q[\sqrt{-5}]$. And if so why are they in ...
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0answers
37 views

An ideal in a ring of polynomials and a field extension.

Let $K\subseteq L$ be fields and $I$ an ideal of $K[x_1,...,x_n]$. I want to show that $IL[x_1,...,x_n]\cap K[x_1,...,x_n] =I$. The inclusion $I \subseteq IL[x_1,...,x_n]\cap K[x_1,...,x_n]$ is ...
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1answer
70 views

Do we really need Zorn's lemma to prove the existence of prime ideals?

Let $A$ be a ring, $S$ a multiplicative set, and $I$ an ideal of $A$ disjoint from $S$; then there exists a prime ideal $P$ of $A$ containing $I$ and disjoint from $S$. The author of the book ...
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0answers
17 views

Splitting field of a cubic polynomial understanding

The cubic polynomial $f(x) = x^3+px+q\in K[x]$ has 3 roots $a_1,a_2,a_3\in \mathbb C$ Hence, the splitting field extension $L=K(a_1,a_2,a_3)$ $\delta=(a_1-a_2)(a_1-a_3)(a_2-a_3)\in L$ since ...
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1answer
15 views

F=F3[x]/(x3+2x-1) where F3 is the field with 3 elements then which of the following are correct

$F=\mathbf F_3[x]/(x^3+2x-1)$ where $\mathbf F_3$ is the field with $3$ elements, then which of the following are correct? $F$ is a field with $27$ elements. $F$ is a separable, but not a normal ...
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2answers
37 views

On the existence of finitely generated modules with finite injective dimension

Assume $R$ is a commutative local Noetherian ring. It is known that if there is a finitely generated module with finite injective dimension then $R$ is Cohen-Macaulay. My question is: if $R$ is ...
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0answers
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Generic freeness (a Lemma from Matsumura, CRT)

Let $B$ be a Noetherian ring, and $C$ a $B$-algebra generated over $B$ by a single element $x$; let $E$ be a finite $C$-module, and $F\subset E$ a finite $B$-module and $CF=E$. Then $D=E/F$ has an ...
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3answers
42 views

Prove that $(0)$ is a radical ideal in $\mathbb{Z}/n\mathbb{Z}$ iff $n$ is square free

Let $n>1$ be an integer. Prove that $0$ is a radical ideal in $\mathbb{Z}/n\mathbb{Z}$ if and only if $n$ is a product of distinct primes to the first power (i.e., $n$ is square free). Deduce that ...
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2answers
51 views

Why is F[x] a UFD? [duplicate]

When reading the proof for if $R$ is a UFD, then $R[x]$ is a UFD, the author uses a fact that $F[x]$ is a UFD. I don't quite understand this. Why $F[x]$ is a UFD? ($F$ is the fraction field of ...
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0answers
60 views

Let $R$ be a ring. Is it true that if $(ab)^n = ab$ for all $a,b \in R$, then $R$ is commutative?

Let $R$ be a ring. Is it true that if $(ab)^n = ab$ for some $n >0 $, for all $a,b \in R$, then $R$ is commutative? Suppose $R$ is a ring and $\exists n \in \mathbb{Z}_{> 0}$ such that ...
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1answer
32 views

polynomial inverse in rings understanding

This problem and solution are in the book. I need help understanding the solution. Problem: Let u be a root of the polynomial $x^3+3x+3$. In $\mathbb Q(u)$, express $(7-2u+u^2)^{-1}$ in the form $a ...
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0answers
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Linear equations in non- commutative rings

Please any reference about general solutions of simple equations of the form $ax=b, xa=b, axb=c$ over non-commutative rings
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1answer
45 views

Boolean Algebra: Is this equality or inequality?

Consider: $$xy + x'y' + yz = xy + x'y' +x'z$$ Is this equality true? I know I could a truth-table but I prefer doing it algebraically. I think there's something tricky here (Like adding a term, ...
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3answers
53 views

Use the the division algorithm and the fact that $I= (\alpha)$ [closed]

Prove that the quotient ring $Z[i]/I$ is finite for any non zero ideal $I$ of $Z[i]$.
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1answer
26 views

In this proof, why is $\gamma[N]$ a proper subset of R/M?

I have highlighted what I do not get in red: I need that $\gamma[N]$ is a proper subset of $R/M$. Why is it that we have this? I get that $\{0+M\}$ is a proper subset of $\gamma[N]$, since the ...
2
votes
1answer
71 views

Is a surjective $R$-endomorphism over a finitely generated $R$ algebra always bijective?

Let $R$ be a unital commutative ring and $A$ a finitely generated $R$-algebra. I found out that if $R$ is a field, then any surjective $R$-endomorphism over $A$ must be injective, too. Does that hold ...
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2answers
28 views

Irreducible in rings understanding

Problem: Let $a$ be an irreducible element of a principal ideal domain $\mathbb{R}$. If $b\in\mathbb{R}$ and $a$ does not divide $b$, show that $a$ and $b$ are relatively prime My attempt: let ...
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1answer
27 views

Right ideals in a ring

Could anyone provide me with an easy presentation of the specifics of a right ideal in a ring (inclusiveness properties, whatever)? Keep it relatively simple and water-clear, please :) Thanks in ...
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1answer
34 views

Group ring C[Z/n] and Artin-Wedderburn decomposition

I am trying to answer the following questions, which I assume follow on from eachother each other; Write $\mathbb C$[$\mathbb Z$/n] as a product of simple rings. For abelian groups $G_1$, $G_2$, ...
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0answers
25 views

On units in subrings ( or ideal ) and quotient ring of ring with unity

Let $R$ be a finite ring with unity and $S$ be an ideal ( or subring ) , let $R^*$ be the group of units of $R$ and $S^*:=R^* \cap S$ , then does $|S^*|$ divide $R^*$ ? Moreover , if $S$ is an ideal ...
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1answer
15 views

To Show that the same correspondence need not yield a ring homomorphism if $n$ does not divide $m$.

Suppose that $n$ divides $m$ and that $a$ is an idempotent of $Z_n$ (that is, $a^2 = a$). To show that the mapping $f(x) = ax$ is a ring homomorphism from $Z_m \to Z_n$. To Show that the same ...
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1answer
32 views

In the general number field sieve, do we need to know whether powers of elements in the algebraic factor base divide an element $a+b\theta$?

I'm reading this paper trying to implement the number field sieve. http://citeseerx.ist.psu.edu/viewdoc/download?rep=rep1&type=pdf&doi=10.1.1.219.2389 Let $\theta$ be the root of some monic ...
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1answer
34 views

Find all units in the ring R as defined.

R = Z[sqrt(-7)] =< C In other words, R is the set of all integers and integers multiplied by the square root of -7. I believe this might be called Z adjoin root -7 but I am not completely sure. ...
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1answer
63 views

The elementary symmetric functions are a homogeneous system of parameters for the invariant ring of a permutation representation

For a permutation representation of order $n$ over a field $F$, I need to show that the elementary symmetric functions $s_1,s_2,\ldots,s_n$ form a homogeneous system of parameters for the ring of ...
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3answers
31 views

Ring Isomorphisms

Why are $\mathbb{Z} [\sqrt{2}]$ and $\mathbb{Z} [ \sqrt {2} ] \times \mathbb{Z} [ \sqrt {2} ]$ not isomorphic to each other? Put this question off for like a week while studying but I feel ...
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2answers
43 views

Show $\Bbb R$ and $\Bbb R[X]$ not isomorphic

Why are the rings $\mathbb{R}$ and $\mathbb{R}[ x ]$ not isomorphic to eachother ? Think it might have to do with multiplicative inverses but I'm not sure.
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2answers
28 views

Why in a graded ring $A$ finitely generated that's an algebra over a field $K$ every maximal ideal is a $K$-subspace?

Probably this question has already been asked, but I'm very bad in find old question and I searched for half an hour, so I'm asking it again. I suppose that's true beacuse my professor used this ...
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0answers
19 views

Condition for an integer prime to be a Gaussian prime

I have a basic question: To show that an integer prime $p$ is a Gaussian prime (i.e. a prime in the ring of Gaussian integers $\mathbb Z[i]$) if and only if the equation $x^2+y^2=p$ has no integer ...
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0answers
24 views

Example of a factorization into irreducible elements that is not unique? [duplicate]

I was reading about factorization into irreducible elements but could not think of any cases off the top of my head where the factorization would not be unique. Could you share some examples? Thanks!
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2answers
17 views

Boolean Algebra - reducing a function

Let $$f(w,x,y,z) = w'x'y'z' + w'x'yz' + wx'yz'$$ How can you reduce it to: $$x'z'(w' +y)$$ Thanks!
2
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1answer
89 views

Definition of multiplicity

Q.1. Bruns_Herzog define multiplicity (in the case of graded rings and modules) as My question is that: why multiplicity for $d=0$ it is defined as $\ell(M)$? Is there a kind of ...