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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How to establish this as being true or false?

Let $R$ and $R^\prime$ be rings such that $R$ has unity $1$ and $R^\prime$ has no $0$ divisors; let $\phi \colon R \to R^\prime$ be a homomorphism such that $\phi [ R ] \neq \{ 0^\prime\}$, where ...
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218 views

Homomorphic Compression

Can there be an algorithm such that, given plaintext data P,Q, and compression function e, Such that if we treat P and Q as a number (a series of bits): $$\begin{eqnarray*}e(P + Q)& =& e(P) ...
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4answers
83 views

Consider the ideal $\langle 1+i \rangle$ in $\mathbb{Z}[i]$.

Consider the ideal $\langle 1+i \rangle$ in $\mathbb{Z}[i]$. $(a)$ Make use of the given description of this ideal, $\hspace{75pt}$ $\langle 1+i \rangle = \{a+bi:a+b \text{ is even}\}=\{\alpha\in ...
3
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3answers
256 views

Isomorphisms and the Fundamental Homomorphism Theorem

Let $$ R=\left\{ \begin{bmatrix} a & b \\ 0 & a \end{bmatrix} : a,b∈ℝ\right\}⊂M_2(ℝ) $$ and $$ I=\left\{ \begin{bmatrix} 0 & b \\ 0 & 0 \end{bmatrix}: b∈ℝ\right\}. $$ Identify the ...
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1answer
58 views

Isomorphism- product of ideals

If $A$ and $B$ are two rings,and $\alpha$ is an ideal of A and $\beta$ is an ideal of B, then $\alpha \times \beta $ is an ideal of $A \times B$. I have to prove that $A \times B / \alpha \times ...
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1answer
49 views

Abstract Algebra (Ring Homomorphisms and Ideals)

Show that the equation $y^2=4$ has at least $4$ solutions in the ring $\mathbb{Z}/5[x]/\langle x^2+1\rangle$. What do you conclude? My main question about this is what $\mathbb{Z}/5[x]/\langle ...
2
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2answers
78 views

A Ring of Square Roots

$\forall x \exists y(x = y \cdot y)$ is true for the trivial ring and $\mathbb{Z} /2 \mathbb{Z}$. Is it true in any other $\mathbb{Z} /n \mathbb{Z}$ rings?
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2answers
90 views

Tensor product of a ring with itself

If $R$ is a commutative ring then $R \otimes_{R} R \cong R$. Is this still true if $R$ is non-commutative?
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1answer
120 views

Krull dimension of this local ring

I want to know what the Krull dimension of this ring $\mathbb C[x,y]_p/(y^2-x^7,y^5-x^3)$ is, where $p\neq (0,0)$. I know the dimension of it in the origin point, but I don't know other cases.
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1answer
129 views

What is the Krull dimension of this local ring

I want to know what is the dimension of this ring $\mathbb C[x,y]_{(0,0)}/(y^2-x^7,y^5-x^3)$. I don't know how to do that. If I suppose $y^2=x^7$ I will get a higher degree of $x$.
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2answers
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Question on decomposing a noetherian ring into product of PIDs

Let $R$ be a reduced noetherian rings of dimension $d$, $S$ be a multiplicative set of all regular elements of $R$, and $K=S^{-1}R$ be localisation of $R$. Show that $K[T]$ (polynomial in 1 ...
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1answer
34 views

Decide the dimension of maximal ideals

Let $A=\mathbb C[x,y]/(y^2-x^3,y^5-x^3)$. I want to know the dimension of each maximal ideal over $\mathbb C$. Actually I can't decide it's maximal ideal. And how to decide its dimension?
2
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1answer
380 views

Ring of continuous functions on $\mathbb{R}$, maximal ideal, quotient

Let $I(S) = \{f \in \mathcal{C}(\mathbb{R}) \ | \ \ \forall x \in S: f(x)=0\}$ I've already proven that it is an ideal in the ring $\mathcal{C}(\mathbb{R})$. However, I have troubles proving that ...
0
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1answer
77 views

The maximal ideals in the polynomial ring

Let $A=\Bbb C[x,y]/(xy,y(y-a))$. I want to know the maximal ideals in $A$. I don't know how do deal with it, I'm confused by the structure of $A$.
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0answers
140 views

First Order Definitions of Finite

I would like some predicates in the language of first order Peano arithemetic (PA) that are true for the standard natural numbers and false for other types of numbers like negative numbers, fractions, ...
2
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0answers
68 views

homework: rings, matrices and polynomials

$A,B$ are both $n \times n$ and diagonal matrices. Prove that there is a matrix $X$ which is $n \times n$, and polynomials $p$ and $q$ such that $A= p(X), B= q(X)$ Is this true for ANY 2 matrices (we ...
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1answer
86 views

homework about homomorphisms : find all the homomorphisms

Find all the continuous homomorphisms $T:\mathbb{R} \rightarrow \mathbb{R}$ Find all the homomorphisms $T:\mathbb{C} \rightarrow \mathbb{C}$ (complex field) such that $T(x)=x$ for every $x$. ...
2
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1answer
85 views

Simple ring and field

We know that the center of a simple ring with unity is a field. But I couldn't make an example of a ring which is not simple but its center is a field. Is it possible? Please give a hint.
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0answers
156 views

Subring with maximal ideals (prime avoidance). Proof verifying and small question

Let $t∈\Bbb N$ and let $p_1, \dots ,p_t$ be $t$ distinct prime numbers. Show that $$R = \{α∈\Bbb Q : α = m/n \mbox{ for some } m ∈ \Bbb Z \mbox{ and } n∈\Bbb N \mbox{ such that } n \mbox{ is ...
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1answer
77 views

Is $[0,\infty)$ the maximum candidate in $\mathbb{R}$ for the positive numbers?

Let us regard $0$ as a positive number, at least for this question, and define that a subset $S$ of a ring $R$ is a candidate for the positive numbers, or simply a "candidate," iff $S$ is a ...
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1answer
93 views

Some basic questions about matrix rings and reversibility.

Neither commutative rings nor division rings are viable approaches to studying rings of matrices. However, there is a very cool notion of a reversible ring, which looks like it can fill this void. I ...
2
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1answer
38 views

Does there exist a reversible monoid that fails to be Dedekind-finite?

Call a ring with unity reversible iff $xy = 0$ implies $yx = 0$. Dedekind-finite iff $xy = 1$ implies $yx = 1$. It is proved here that every reversible ring is Dedekind-finite. Now clearly, the ...
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3answers
290 views

How many irreducible monic quadratic polynomials are there in $\mathbb{F}_p[X]$?

Can some of you help me with my homework? I had to count the irreducible quadratic, monic polynomials in $\mathbb{F}_p[X]$ for arbitrary p. I will show you what I tried myself. Research effort First ...
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1answer
70 views

On the extension of fields

Let $F\subseteq K$ be a finite field extension and let $a_1,..., a_n$ be an $F$-basis for $K$. I want to show that the matrix $A := (tr(a_ia_j))$ is singular if and only if $tr K =0$. Any suggestion ...
1
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1answer
121 views

When each prime ideal is maximal [duplicate]

By $A$ we denote an arbitrary commutative ring. Suppose that for each $a\in A$ $(a)=(a^2)$ (i.e. $\forall a\in A$ $\exists b$ such that $ba^2=a$). It is easy to see that this implies that each prime ...
4
votes
2answers
60 views

Ring-Homomorphism from $\mathbb{Z}_{2}$ to $\mathbb{Z}_{2n}$

Let $n$ be a positive integer. Then the problem is to show that there is a ring-homomorphism from $\mathbb{Z}_{2}$ to $\mathbb{Z}_{2n}$ if and only if $n$ is odd. My effort : let $\phi$ be such a ...
1
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1answer
109 views

Problem with structure of a semisimple ring theorem

Structure of semisimple ring (Wedderburn-Artin) in Rings and Categories of Modules - Frank W. Anderson, Kent R. Fuller (auth.) Proof: Please explain that: "Now $_RR$ is direct sum off these ...
3
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1answer
56 views

Show that all homomorphsms phi are of the same form (problem)

Problem: Let $p$ be an odd prime. Let $G = Z_p \times Z_p$. a group under addition. (Ignore multiplication in this problem.) Let $a,b,c,d\in Z_p$ and define $\phi: G\rightarrow G$ by $\phi((i,j)) = ...
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0answers
140 views

Show a set of polynomial functions is a ring.

Let $T$ be the set of all polynomial functions from $\Bbb{Z}_3 \to \Bbb{Z}_3$. Show that $T$ is a ring. Show that $T$ is a finite ring with zero divisors. (Hint: consider $x + 1$ and $x^2 + 2x$.) Show ...
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1answer
71 views

Homework basic abstract algebra

My question is as follows: $R$ is a ring such that for all $x \in R, x^2=x$ $p$ is a prime ideal of $R$. Show that $R/p$ (R modulu p) has exactly 2 elements. What I did: $x^2=x$ $x^2-x=0$ ...
3
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2answers
131 views

The pre-image of an irreducible element.

Problem: Let $ R,S $ be integral domains and $ f: R \to S $ a unit-preserving homomorphism. Assume that $ x \in S $ is irreducible. Then does the pre-image $ {f^{-1}}[\{ x \}] $ contain only ...
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1answer
51 views

Proof of Char(R)

Suppose we have a ring $R$ such that $char(R)$=$k$ with $k>0$. (a) Let $p=mk$, where $m\in \mathbb{Z}$. Prove that $px=0_R$for all $x\in R$. [Remember that $p$ is an integer and $x$ is an element ...
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0answers
42 views

Minimal spectrum of graded rings

Let $R$ be a left Noetherian ($\mathbb N$-)graded ring and let $R_0$ be its $0$-th component. When $R$ is commutative it is well-known, and easy to prove, that the minimal prime ideals of $R$ are ...
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1answer
134 views

How to prove that every simple left $R$-module is isomorphic to a minimal left ideal of $R$

We know that: $T$ is a simple left $R$-module $\Longleftrightarrow T\cong R/M$, where $M$ is a maximal left ideal of $R$. So please tell me how to prove that every simple left $R$-module is ...
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1answer
165 views

Order of an element in a finite cyclic group

Let $G$ be a cyclic group of order $m$ generated by an element $a$. I want to show that the order of $a^k$ is $m/d$, where $d:=\gcd(k,m)$. I have a simple proof, but want to make sure I haven't ...
2
votes
1answer
236 views

Ring homomorphisms map units to units

Let $R$, $S$ be commutative rings, and let $\Phi:R\to S$ be a homomorphism of rings. Prove that if $a\in R$ is a unit, then $\Phi(a^{-1})=\Phi(a)^{-1}$. Deduce that $\Phi$ maps units of $R$ to units ...
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2answers
62 views

solutions of an equation over a quotient ring

so far i can see that it has 3 solutions but im not sure where to find the others that the question hints at. Show that the equation $y^2=4$ has at least four solutions in the ring $Z_5[x]/\langle ...
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2answers
152 views

Is a vector space a ring, integral domain or field?

Is a vector space a ring, integral domain or field, with respect to scalar multiplication? If you could give me an example, that would be awesome!
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1answer
127 views

Prove that $\mathbb{Z}\left[\sqrt{-3}\right]$ is not a Dedekind domain. [closed]

Prove that $\mathbb{Z}\left[\sqrt{-3}\right]$ is not a Dedekind domain.
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4answers
342 views

Are there non commutative rings with no zero divisors?

If there are, Are there unity (but not division) rings of this kind? Are there non-unity rings of this kind? Sorry, I forgot writting the non division condition.
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3answers
78 views

Sum of Products of Ideals with Itself

Let $R$ be a commutative unital ring and let $I$, $J$ be ideals in $R$ such that $I+J=R$. Show that $II+JJ=R$. What I've tried: Clearly $II+JJ \subseteq I+J =R$ as $I$ and $J$ are closed under ...
0
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1answer
108 views

Find ring homorphisms from $\Bbb Z_2$ to $\Bbb Z_6$

I am trying to find all ring homomorphisms $\phi: \Bbb Z_2\to \Bbb Z_6$. I know this is an easy problem , but I am having trouble grasping it so I cannot find what the answer is. Could someone please ...
2
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1answer
302 views

Projective Modules, Annihilators, and Idempotents

Let $Ra$ be the left ideal of a ring $R$ generated by an element $a \in R$. Show that $Ra$ is a projective left $R$-module if and only if the left annihilator of a, $\{r \in R | ra = 0\}$ is of the ...
2
votes
1answer
86 views

Problem with semisimple ring theorem

Proposition: For a ring $R$ the following statements are equivalent: (a) $R$ has a simple left generator; (b) $R$ is simple left artinian; (c) For some simple $_RT, _RR \cong ...
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2answers
53 views

Proving that $annM$ is a prime ideal for ireducible $M_R$

If we let $R$ be a ring with $1$ and $M_R$ to be an irreducible right $R$-module then I want to show that $annM]\{r\in R|Mr=0\}$ is a prime ideal. So if we take $a,b\in R$ with $aRb\subset annM$ then ...
2
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1answer
91 views

How do I show this is a homomorphism?

Let $F$ be a field, let $R_1 = F[x]$ be the ring of polynomials with coefficients in $F$, and let $R_2$ be the ring of all functions from $F$ to itself, with addition and multiplication defined as the ...
2
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1answer
81 views

The definition of the crossed product of a ring and a monoid

I know the definition of crossed product of a ring and a group. But when I consider the definition of the crossed product of a ring and a monoid, I can not understand exactly the definition on page ...
2
votes
1answer
116 views

How to verbalize $R[x]$?

Let $R$ be a ring, and let $x$ be an indeterminate. Let $R[x]$ denote the ring of polynomials in $x$ with co-efficients in $R$. How to most efficiently read (i.e. pronounce) the symbol $R[x]$ while ...
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1answer
135 views

Valuation Ring with value group $\Gamma=\mathbb{Z} \oplus \mathbb{Z}$

Let $\Gamma=\mathbb{Z}\oplus \mathbb{Z}$ be the free abelian group with two generators, lexicographically ordered. That is, $(a,b)\geq (a^{'},b^{'})$ iff either $a>a^{'}$ or $a=a^{'} \mathrm{and}\, ...
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2answers
2k views

Ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z[\sqrt{-5}]$

Show that the ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z[\sqrt{-5}]$. I fail to understand how can 3 and $1+\sqrt{-5}$ generate an ideal. ...