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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Kaplansky characterization of principal Artin ring

I would like to learn the proof of this paper of Kaplansky where it is proven that for a commutative ring every module split as sum of cyclic module iff the ring is an Artin principal ideal ring (well ...
3
votes
0answers
33 views

Unit group of an imaginary quadratic ring

Let $R$ be an imaginary quadratic ring. Then, the unit group $R^{\times}$ is finite. To prove this, I worked with normal forms, algebraic integers and the fact that $R \not \subset \mathbb{R}$. But I ...
4
votes
3answers
85 views

For a prime integer $p \in \{2, 3, 5, \cdots\}$, is $pR$ a maximal ideal in $R$?

If $R$ is a commutative ring with unit and $p$ is a prime number ($2,3,5,\cdots$), then is $pR$ a maximal ideal of $R$? If not what conditions should I impose on $R$?
3
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1answer
54 views

Prove that $f_n=-1+\prod_{i=1}^{n}(X-i)$ is irreducible in $\mathbb{Z}[X]$

Prove that, for all $n\in \mathbb{N}$, $f_n=-1+\prod_{i=1}^{n}(X-i)$ is irreducible in $\mathbb{Z}[X]$.
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2answers
73 views

Proving the Ideal Generated by the Coefficients of $f(X)\cdot g(X)\in R[X]$ is $R$.

Let $R$ be a commutative ring with unity, and let $f(X),g(X)\in R[X]$. Assume the ideals generated by the coefficients of $f(X),g(X)$ are both $R$. Prove that the ideal generated by the ...
1
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1answer
59 views

If $[A,A]A[\lambda,A] = 0$ then $\lambda \in Z(A).$

Suppose that $A$ is a unital ring and $([A,A]) = A.$ If $[A,A]A[\lambda,A] = 0$ prove that $\lambda \in Z(A).$ Comments: This is part of an exercise I'm doing, I'm posting this part because I am ...
3
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0answers
24 views

Module endomorphisms with the same kernel

Let $R$ be a finite commutative principal ideal ring. Let $n$ be a positive integer. For $i=1, \ldots, n-1$ we let \begin{align*} w_i := w_i(x_{i+1}, \ldots, x_n) = \sum_{j=i+1}^n t_{ij}x_j \in ...
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0answers
8 views

Reference for a Dickson Determinant Polynomial

For $2\leq \ell \leq k$, consider the polynomial \begin{equation} P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k] \end{equation} ...
2
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2answers
182 views

Divisors of zero in polynomial ring

I have the following theorem: McCoy: Let $R$ be a commutative ring with identity. If $f=\sum_{i=0}^na_iX^i$ is a zero divisor in $R[X]$, then there exists a nonzero $c$ in $R$ such that $cf=0$. ...
0
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0answers
31 views

Decomposition and valuation rings

I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theory: $(K,u)$ be a field and its absolute value, $(K_u,\bar u)$ be its completion and absolute value ...
2
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1answer
30 views

Modify this formula : $R/I \cong \phi[R]/\phi[I]$

Let $R$ be a ring and $I$ an ideal of $R$, and let $\phi : R\longrightarrow R'$ be a ring homomorphism. Studying by myself, I have a conjecture the following: $$R/I \cong \phi[R]/\phi[I].$$ This ...
2
votes
1answer
25 views

Prove that $R/(2i)$ and $\mathbb{Z}/4\mathbb{Z}$ are isomorphic rings

Let $R=\{a+2ib|a,b\text{ integers}\}$.Prove that $R/(2i)$ is isomorphic to $\mathbb{Z}/4\mathbb{Z}$
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0answers
15 views

Local Constancy of Rank Function

Recently I asked this question. I believe that I have come up with a solution, but I am unsure, because the proof I have seems too easy to be true, and doesn't make very many assumptions. My ...
0
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1answer
51 views

Endomorphisms of $\mathbb{Z}/n\mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$

Let G be any abelian group, End(G) be the set of all group homomorphisms $\varphi\colon G\to G $. End(G) is a unital ring under the operations + and $\cdot$(Please refer to the link for detail, ...
0
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1answer
38 views

Every prime ideal is maximal [duplicate]

Problem: Show that if R is a finite ring, then every prime ideal of R is maximal. My attempt: Let I be a prime ideal of R. Then, by definition of a prime ideal, ab ∈ I implies a ∈ I or b ∈ I for ...
4
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1answer
91 views

finitely generated ideal and number of generators

let $I$ be a finitely generated ideal of ring $R$. Suppose $I/I^2$ as an $R/I$ module is generated by $r$ elements then question is to prove that $I$ is generated by $r+1$ elements.. I have tried ...
2
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1answer
21 views

Is there a binary operator (besides composition) closed under permutations or a notion of a metric space on permutations?

When i say "a binary operator closed under permutations" I mean, given $2$ (finite, same number of elements) permutations $p_1$, $p_2$ , is there an operator "$+$" such that $p_1+p_2=p_3$ ($p_3$ a ...
7
votes
1answer
71 views

Is this ring a PID? [closed]

Let $R$ be the $k$-subalgebra of $k(t)$ generated by the set $k[t]$, of all polynomials, and a pair of rational functions: ${1\over{t-1}}$ and ${1\over{t-2}}$. Is the ring $R$ a PID?
1
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1answer
39 views

Identity mapping

Problem: Show that the identity mapping is the only ring homomorphism from $\mathbb{Z}$ to $\mathbb{Z}$. Solution: Let $\varphi:\mathbb{Z}\rightarrow\mathbb{Z}$ be a ring homomorphism. Then ...
1
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1answer
53 views

Ring homomorphism

Problem: Let $R$ be a commutative ring, and let $D$ be an integral domain. Let $φ : R → D$ be a nonzero function such that $φ(a+b) = φ(a) + φ(b)$ and $φ(ab) = φ(a)φ(b)$ for all $a,b \in R$. Show ...
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0answers
31 views

Prove that $E(x)y + E(y)x \in Z(A)$ for all $x,y \in A$.

Let $A = M_2(C)$, where $C$ is any commutative ring. Define $E: A \longrightarrow A$ by $E(x) = x - tr(x)Id_2$, where $tr(x)$ denotes the trace of $x$. Prove that $E(x)y + E(y)x \in Z(A)$ for all $x,y ...
0
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1answer
38 views

Show that $([A,A])$ contains the identity matrix.

Suppose $A = M_n(C)$, $n \geq 1$, where $C$ is a commutative unital ring. If $n \geq 2$, then $([A,A])$, the ideal generated by $[A,A]$, contains the identity matrix.
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1answer
38 views

characteristic of a ring

I got 4 short question about characteristic. 1) What is characteristic of integral domain D which suffices $20 \cdot 1_D=0_D=12 \cdot 1_D$ 2) Let $A=\{0,1,a\}$ be a integral domain what is ...
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0answers
33 views

Existence of a splitting ring

Let $R$ be a commutative ring and $f\in R[X]$ be a monic non-constant polynomial. How can one show that there exists a commutative ring $S$ so that $R$ is a subring of $S$ and $f$ can be written as a ...
3
votes
1answer
47 views

Adjoining an identity to a ring

I am run into the following in an Algebra text: "Let $R_0=\mathbb Z/2\mathbb Z⊕\mathbb Z/2\mathbb Z⊕\cdots$ viewed as a ring without identity, with addition and multiplication defined componentwise. ...
1
vote
1answer
30 views

Gauss lemma for arbitrary commutative ring [duplicate]

Part (iv) of exercise #2 for chapter 1 in Atiyah and Macdonald's book Introduction to Commutative Algebra asserts that if $f, g \in A[x]$ are primitive then $fg$ is primitive. We know that this is ...
0
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1answer
35 views

Minimal polynomial is irreducible

Suppose $\mathbb{E}$ is a field extension of $\mathbb{F}$. If $a$ is algebraic over $\mathbb{F}$ we define the minimal polynomial for $a$ as the monic irreducible generator $g$ of the ideal ...
1
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1answer
48 views

Multiplicative inverse of the power series $e^x - c$ for $c \neq 1$.

We know that the power series $f(x)= e^x -c \in \mathbb C[[x]]$ for $c \neq 1$, has a multiplicative inverse, since it's constant coefficient is non-zero. I was wondering whether the inverse is known ...
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0answers
22 views

Prime ideals in $C(X)$ and $C^*(X)$ and to be correspond

we know that every maximal ideal in $C(X)$ is in this form: $$M^p=\left\{\,f \in C^*(x):\ p\in cl_{\beta X} Z\left(f\right)\,\right\}$$ and every maximal ideal in $C^*(X)$ is ...
4
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1answer
34 views

Perfect number in gaussian integers

We have complete description about irreducibles in the ring Z[i],of gaussian integers. Now I was trying to define suitably the notion of "perfect number" in Z[i]. But the problem is unique ...
1
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0answers
13 views

Can We Use Extension of Scalars (Or Some Other Neat Way) to Prove This

Let $k$ be a field and $b(x)$ be a polynomial of degree at least $1$ in $k[x]$. Then for any given $f(x)\in k[x]$, there is an integer $m$ and polynomials $d_0(x), \ldots, d_m(x)\in k[x]$ such that ...
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votes
2answers
100 views

Prove that a finite Boolean ring is isomorphic to $\mathbb Z_2\times \mathbb Z_2\times\cdots\times \mathbb Z_2$ [duplicate]

If $R$ is a finite Boolean ring with $1\neq 0$ then prove that $R\cong \mathbb Z_2\times \mathbb Z_2\times \cdots\times \mathbb Z_2$. How should I proceed? Please give some hints only.
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2answers
62 views

Jacobson radical of $\mathbb{F}_{2}(t)[x]/(x^4-t^2)$

Let $\mathbb{F}_{2}$ be the field of two elements. Let $R=\mathbb{F}_{2}(t)[x]/(x^4-t^2)$. Why is $R/J(R)$ equal to $\mathbb{F}_{2}(t)[x]/(t-x^2)$? here $J(R)$ denotes the Jacobson radical of $R$.
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1answer
43 views

Proving $\gcd(f_i)=1\Rightarrow \mathbb{A}_\mathbb{C}^n\setminus \{f_i\}$ is not affine

I need to prove the following lemma: Lemma: Let $f_i\in \mathbb{C}[x_1,\dots,x_m]$ s.t. $\gcd(f_1,f_2,\dots,f_n)=1\quad(1<n\le m)$. Prove that the variety ...
3
votes
1answer
75 views

Is there any proper subring of $\mathbb{R}$ with field of fractions equal to $\mathbb{R}$?

Is there any proper subring of $\mathbb{R}$ with field of fractions equal to $\mathbb{R}$? Can we construct that proper subring? Is it necessarily an integral domain?
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1answer
90 views

How do modern algebraists think about diagonal matrices?

Let $\mathbb{K}$ denote a field and $A$ denote a $\mathbb{K}$-algebra. Then given a $\mathbb{K}$-subalgebra $\Delta$ of $A$, I suppose it make sense to declare that $m \in A$ is ...
1
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1answer
40 views

Unit cancellation in group rings

Suppose I have a finite group $G$, a non-trivial proper subgroup $H$, a field $k$ (restricting to $k=\mathbb C$ would be fine), and non-zero elements $a,u$ in the group algebra $kG$ satisfying the ...
3
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1answer
42 views

How to show that an ideal is maximal

How do you show that $\langle y^2+2, x-1 \rangle$ is a maximal ideal in $\Bbb Q[x,y]$? I know that if you add another element that is not in this ideal, you should get the whole ring, thus ...
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0answers
49 views

Determining prime ideals lying above a given ideal

Let $R=\mathbb{Z}[x]/(f)$, where $$f(x)=x^4+42x^3-11x^2+22x-2002002002002002.$$ Let $I=3R$, the ideal generated by $3$ in $R$. Find all prime ideals of $R$ that contain $I$. I am hoping to ...
2
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1answer
31 views

Question on finitely generated algebras [duplicate]

While reading Reid's "Introduction to Algebraic Geometry", I came across the following passage: "A finitely generated $k$-algebra is a ring of the form $A = k[a_1,\cdots,a_n]$, so that $A$ is ...
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0answers
27 views

Proof review: Every maximal ideal of ring of continuous functions has the same form

Let $R$ be the ring of real-valued continuous functions on $[0,1]$. If $M$ is a maximal ideal of $R$ prove $\exists \lambda \in [0,1]$ s.t. $M = \{f(x) \in R : f(\lambda) = 0 \}$. (from Herstein ...
3
votes
3answers
34 views

Checking whether $x^2-5$ is prime but not maximal

I want to find an example of a prime ideal that is not maximal. I thought about $x^2-5$. We know that $Z[\sqrt{-5}]\cong Z[x]/(x^2-5)$ is an integral domain, therefore is $x^2-5$ prime. However, I ...
0
votes
2answers
35 views

Ring of rational 2x2 matrices has no proper ideals [duplicate]

I'm trying to prove the ring $R$ of all rational 2x2 matrices has no (two-sided) ideals other than $(0)$ and $R$. My attempt: Let $I$ be an ideal other than $(0)$. Then there exists some nonzero 2x2 ...
2
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2answers
17 views

Irreducibility criteria for polynomials with several variables.

Let $K$ be a field. Show that $x^2-yz$ is irreducible in $K[x,y,z]$. Deduce that $x^2-yz$ is prime. If it is $K[x]$, then there are several methods which can be used to check whether a given ...
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2answers
99 views
2
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2answers
54 views

What is the quotient ring $\mathbb{Z}_2[x,y]/(x+y)$?

Okay, so $\mathbb{Z}_2[x,y]/(x)\cong\mathbb{Z}_2[y]$, that much I see. But what is $\mathbb{Z}_2[x,y]/(x+y)$? I think this should also be a polynomial ring in one variable, is that true? Please help, ...
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0answers
57 views

Constant Dimension for Localization of Projective Modules

It is a well known fact that the localization of a projective module over a commutative ring is free. However, I don't know anything about the dynamics of how the dimension of the resultant free ...
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1answer
214 views

Proof a Rng cannot have exactly five non-zero divisors.

Let $R$ be a Rng (a ring which does not necessarily have a $1$). We call an element $a$ regular if $xa=0$ implies $x=0$ and $ax=0$ implies $x=0$. Prove $R$ cannot have exactly five regular elements. ...
7
votes
1answer
59 views

Proving $\{a+b\sqrt{-7}, a,b\in\frac{\mathbb{Z}}{2}\}$ is a euclidean domain

I am having a bit of difficulty showing that the set $A=\{a+b\sqrt{-7},a,b\in\mathbb{Z}/2\}$ (i.e. using integers+half integers) is a euclidean domain. I see how a lot of the proofs work, like for ...
5
votes
2answers
56 views

How do ideal quotients behave with respect to localization?

Suppose $R$ is commutative ring with unity. For ideals $I$, $J \subseteq R$, the ideal quotient $(J:I)$ is $$(J:I) := \{x\in R \, : \, xI \subseteq J\}$$ Let $S\subset R$ be a multiplicative set. When ...