This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

learn more… | top users | synonyms (1)

2
votes
2answers
45 views

A question on Join homomorphism and Ideals

On page 287 of the book Mathematical Methods in Linguistcs, by Barbara Partee, Alice Ter Meulen and Robert E. Wall (Dordrecht, Kluwer Academic Press, 1993), I find the following theorem, which they ...
2
votes
1answer
133 views

Regular subrings of a polynomial ring

Let $R=\mathbb{C}[x,y]$. I have the following situation: $\mathbb{C} \subseteq D \subseteq R$ is affine (= finitely generated as a $\mathbb{C}$-algebra), noetherian, has field of fractions ...
10
votes
0answers
112 views

Example of a commutative, local, dual ring with nilradical $N$ such that $ann(N)\nsubseteq N$

For an ideal $I\lhd R$ in a commutative ring $R$, let $ann(I)$ denote the annihilator of $\{x\in R\mid xI=\{0\}\}$. A commutative ring $R$ is said to be a dual ring if for every ideal $I$ of $R$, ...
5
votes
1answer
68 views

$R$ is a finite ring and for every $a \in R$, there exists a natural number $n(a)$ such that $a^{n(a)}=a$

$R$ is a finite ring and for every $a\in\,R$ there exist natural number $n(a)>1$ that $a^{n(a)}=a$ . Is $R$ a ring with identity? If this question is correct then, for every $a \in ...
1
vote
0answers
24 views

If $I$ is a finitely generated semisimple left $R-$module, show that $I=Re$ for an idempotent $e \in I$.

Let $I$ be a left ideal in a semiprime ring $R$. If $I$ is a finitely generated semisimple left $R-$module, show that $I=Re$ for an idempotent $e \in I$.
1
vote
1answer
56 views

Regularity of a quotient ring of the polynomial ring in three indeterminates

Let $I=(f)$ be a prime ideal in $R=\mathbb{C}[x,y,z]$, so $f$ is an irreducible polynomial, and further assume that $f$ is of the following form: $f=z^n+c_{n-1}z^{n-1}+\ldots+c_1z+c_0$, where ...
3
votes
1answer
94 views

Question about ideals of a ring: $IJ=I \implies J=I$?

Doing exercises, this question came to my mind. Is it true that if $I$ and $J$ are proper and nonzero ideals of a ring $R$, $$IJ=I \implies I=J?$$ And $$IJ=I \iff I\subseteq J?$$
2
votes
1answer
20 views

Describing the Kernel of an identity-preserving ring morphism from a ring $R$ to an Endomorphism ring of an additive Abelian Group.

I'm currently working through TS Blyth's book on Module Theory (Module Theory: An Approach to Linear Algebra). From Exercise 2.3: "Let $M$ be an Abelian Additive Group, and $R$ a unitary ring. Let ...
1
vote
2answers
68 views

Problem about ideals of the localization of a ring

I'm having problems on doing the section (ii) of this exercise. Let $R$ be a domain. Let $P$ be a prime ideal of $R$. (i) Prove that $S:=R\setminus P$ is a multiplicatively closed system with no ...
3
votes
0answers
106 views

Matrix representation of $\mathbb{C}$ as $^*$Algebra.

We know that there are many matrix representations of the field $\mathbb{C}$. For $2 \times 2$ real entries matrices, e.g., all the subrings of $M(2,\mathbb{R})$ generated by $I$ and a matrix $J$ such ...
2
votes
1answer
68 views

Number of generators of prime ideals in $K[x_1,x_2,…,x_n]$

Is there any bound for the number of generators of prime ideals in $K[x_1,x_2,...,x_n]$? (For example in $K[x,y]$.) We know that maximal ideals of $K[x_1,x_2,...,x_n]$ have $n$ generators.
2
votes
1answer
39 views

$K[x,y]$ (where $K$ is a field) have any bound for the number of generators of ideals?

We know that maximal ideals of $K[x_1,x_2,...,x_n]$ have $n$ generators. But is there any bound for the number of generators of arbitrary ideals? (For example in $K[x,y]$.)
3
votes
1answer
67 views

Nilpotents after tensoring with a field

Let $A \to B$ be a homomorphism of commutative rings with unit. Let $A_{\text{red}}=A/ \sqrt{(0)}$ and $B_{\text{red}}=B/ \sqrt{(0)}$ be the corresponding reduced rings. Now let $A_{\text{red}} \to K$ ...
3
votes
3answers
53 views

In a commutative ring with identity, if $p$ is irreducible, is ($p$) a maximal ideal?

In a Euclidean Domain, $D$, if we mod out by an irreducible, $p$, we get the field $D/(p)$. I can see that this follows since we are going to be able to write $1$ as a linear combination of $p$ and ...
2
votes
2answers
53 views

If $1\in M$, could $M$ be a maximal ideal of a commutative ring with identity $R$?

If $1\in M$, could $M$ be a maximal ideal of a commutative ring with identity $R$? I know that this is a very silly question. I think that the answer is that $M$ can't contain the identity for ...
3
votes
1answer
55 views

Working in $\mathbb Q[x]$. Two polynomials are coprime if their gcd is a constant?

When are two polynomials coprime? Is it when their gcd is a constant? If we divide $x^3-7x-5$ by $x-4$, we get: $$x^3-7x-5=(x-4)(x^2+4x+9)+31$$ So, is $31$ their gcd, but since $31$ is not monic ...
3
votes
2answers
53 views

about center of group rings $RG$ and $(R/I)G$

Let $I$ be an ideal of a ring $R$. It is mentioned in the book An Introduction to Group Rings (by Sehgal and Milies) that the canonical homomorphism $RG \rightarrow (R/I)G$ maps $Z(RG)$, center of ...
8
votes
0answers
81 views

Classification of all subrings

Let $R$ be an integral domain whose underlying additive group is finitely generated free and whose field of fractions $K$ is a finite Galois extension of $\mathbb{Q}$. Is there a method of ...
0
votes
0answers
47 views

Evaluating composition of functors

Let $R$ be a ring and $S$ its $n \times n$ matrix ring. We consider the categories $_R Q$ and $_S Q$ of their respective left modules. We define a functor $F \colon _R M \to _S M$ by $$ F(M) = M^n $$ ...
0
votes
1answer
64 views

How to show $1+\sqrt 2$ generate an infinite cyclic group of units in $\mathbb Z[\sqrt 2]$?

The answers given here seem very convoluted: The units of $\mathbb Z[\sqrt{2}]$. Is it possible to provide a more explanatory proof?
4
votes
4answers
364 views

infinitely many ideals

does the ring $\Bbb Z_2[x]$ have infinitely many ideals like $\Bbb Z[x]$? How do you know if a ring has a finite number of ideal. particularly asking about seemingly large rings.
0
votes
0answers
9 views

finding prime elements in Z[√2 ] [duplicate]

I am trying to find prime elements of z[√2],and I'am trying to have a procedure like finding prime elements in z[i],is it correct or not?? tnx for your help.
1
vote
1answer
62 views

Is $\mathbb{Q}(\sqrt{3})$ in someway related to Quotient ring?

I can't help but notice that they look exactly the same. For example: $\mathbb{Q}(\sqrt{3})$ = $\lbrace p + q\sqrt{3}:p,q \in \mathbb{Q}\rbrace$ That seems pretty much exactly an ideal. Only the ...
1
vote
1answer
72 views

Can we prove, without axiom of choice, that the set of all zero divisors (including $0$) of a commutative ring with unity contains a prime ideal?

Let $R$ be a commutative ring with unity , I know that assuming axiom of choice , if $A$ is the set of all zero divisors (including $0$ ) then it is a union of prime ideals so it contains a prime ...
1
vote
2answers
23 views

Finding the conjugates, why can they argue this way?(exercise)

In one exercise I am supposed to find the conjugate of $\sqrt{2}+i$ over $\mathbb{Q}$. I found the answer by finding irr$( \sqrt{2}+i,\mathbb{Q})$, and then solving the polynomial finding all the ...
4
votes
0answers
61 views

If $\mathfrak{m}\otimes M\rightarrow A\otimes M$ is injective, what else has to be injective?

Let $A$ be a local (not necessarily noetherian) ring with maximal ideal $\mathfrak{m}$ and residue field $k$. Let $M$ be a finitely generated $A$-module such that $\mathfrak{m}\otimes_A M\rightarrow ...
1
vote
1answer
55 views

Find image and kernel of $\varphi: \mathbb{Z}[x] \to \mathbb{C}$ given by $x \mapsto i$

Consider the homomorphism $\varphi: \mathbb{Z}[x] \to \mathbb{C}$ given by $x \mapsto i$. Find: (a) the image; (b) the kernel; (c) exhibit the bijection of the Correspondence Theorem explicitly for ...
2
votes
3answers
87 views

About the group of units

I'm stuck at this section of the following problem: Let $R$ be a commutative ring and $S$ be a subset of $R$, with $S$ multiplicatively closed. Find the group of units of $S^{-1}R$. My try: ...
7
votes
3answers
188 views

Maximal left ideals $\leftrightarrow$ simple left modules

Suppose $R$ is a ring with unity. This passage in Lang's Algebra discusses the correspondence $$\text{Maximal left ideals of $R$} \leftrightarrow \text{Simple left $R$ modules},$$ where I corresponds ...
1
vote
1answer
53 views

Finding primary decompositions of ideals

I have been given this example of the decomposition of an ideal into primary ideals $$ I =⟨x^2,xy,x^2z^2,yz^2⟩$$ Then the primary decomposition of this ideal is: $$⟨x^2,y⟩∩⟨x,z^2⟩⊆K[x,y,z]$$ This ...
1
vote
1answer
33 views

Show that the variety $V(I(X))=X$

In the ring $R=K[x_1,...,x_n]$, the variety of an ideal is defined as $V(I)=\{(a_1,...,a_n)\in K^n|f(a_1,...,a_n)=0, \space\forall f\in I\}$ The ideal of a variety is defined as $I(V)=\{f\in ...
1
vote
0answers
26 views

ring restriction of linear groups

Suppose that there is a group $G\subseteq\text{GL}(n,\mathbb{C})$ defined to be the group satisfying some equations on $\text{GL}(n,\mathbb{C})$(for example, given a nonsingular matrix $A$, ...
0
votes
1answer
30 views

On a question about polynomial ring

Let the ring $ R$ define as the following $R=\{a_1+a_2x^2+a_3x^3+...+a_x^n;a_i\in \mathbb R,\,n\gt 2\}$ and Let the ideal $I$ generated by $<x^2+1,x^3+1>$. Is $I=R$ or not?
3
votes
1answer
57 views

Number of ring homomorphisms form $\mathbb Z[x]$ to $\mathbb Z_{12}$

I have tried : Let $f$ be an homomorphism form $\mathbb Z[x]$ to $\mathbb Z_{12}$. we have to find the possible image of $1$ and $x$. Suppose $f(1) = a$, then $f(1)^2 = f(1) = a^2 = a$, then the ...
0
votes
1answer
38 views

Show the group isomorphism $(\mathbb{Z}/n)^\times \cong (\mathbb{Z}/p_1^{k_1})^\times \times \cdots \times (\mathbb{Z}/p_n^{k_n})^\times$

When $r$ and $s$ are relatively prime we have the ring isomorphism $\mathbb{Z}/rs \cong \mathbb{Z}/r \times \mathbb{Z}/s$ Given a prime factorization of $n$ where $n = p_1^{k_1} \cdots p_n^{k_n}$ ...
1
vote
1answer
34 views

Why is it true that any polynomial $f(x)$ with coefficients in a finite field $F_q$ is separable over that field?

Why is it true that any polynomial $f(x)$ with coefficients in a finite field $F_q$ is separable over that field? I'm confused because the polynomial $x^2+1$ in $F_2[x]$ is inseparable ($f(x)$ and ...
4
votes
1answer
65 views

Determining if $\mathbb{Z}[a]$ is a discrete subring of $\mathbb{C}$.

Let $a \in \mathbb{C}$ and consider the ring $\mathbb{Z}[a]$. Is there some nice criterion which will tell me whether $\mathbb{Z}[a]$ is discrete in the sense that there is some $\delta >0$ such ...
0
votes
2answers
42 views

Prove that $v$ and $\overline{v}$ are not associates in the ring $\mathbb{Z}[i].$

Suppose $p$ be an odd prime such that $p \equiv 1 \pmod 4$. In the ring of Gaussian integers $\mathbb{Z}[i]$, $p$ factors as $p = v \cdot \overline{v}$ for a prime $v \in \mathbb{Z}[i].$ ...
0
votes
1answer
72 views

Show that 2Z and 3Z are not isomorphic - question on proof

I need to show that $2\Bbb Z$ and $3\Bbb Z$ are not isomorphic. I found a contradiction as follows: let $p$ be this isomorphism from $2\Bbb Z$ to $3\Bbb Z$. Then $p(4) = p(2*2) = p(2+2)$, so ...
4
votes
1answer
44 views

For a group ring, finding if a subset is an ideal. [closed]

For the ring $R=SG$, the group ring of a finite group G over an integral domain S, and a subset $I=(g-1|g \in G)$, is this subset an ideal? Is it prime? How about maximal?
1
vote
0answers
23 views

Unit or (Left/right zero) divisior [duplicate]

Let R is finite Ring with 1 and $a \in R \setminus \{0\}$. Show that a is Unit or (Left/right) zero divisior It's obvious that we have to use the mapping: $x \rightarrow ax$ and $x \rightarrow xa$ but ...
1
vote
2answers
81 views

Generalization of Chinese Remainder Theorem

Q: With ring $R$, if $I, J \subseteq R$ are ideals such that $I+J=R$, then the map $R/(I \cap J) \to R/I \times R/J$ given by $a + (I \cap J) \mapsto (a+I, a+J)$ is an isomorphism, broadly ...
2
votes
1answer
20 views

Showing a Ring and an Ideal are equal

Q: Let R be a commutative ring with unity. Prove that if A is an ideal of R and A contains a unit, then A=R. This is my attempt at an answer: It suffices to show that all the elements in R are in A. ...
0
votes
2answers
67 views

Proving something it NOT and integral domain

Let $R$ and $S$ be two commutative rings with unity. Prove that $R\times S$ is NOT an integral domain. This is the best I could think of so far, please give me a push in the right direction and ...
4
votes
2answers
25 views

Given a ring with unity and a central idempotent element e, prove some isomorphic relations

Given a ring $R$ with 1 $\neq$ 0, and an element $e$ that is idempotent and central in $R$, I want to prove that $R/Re \cong R(1-e)$, $R/R(1-e)\cong Re$, and subsequently, $R\cong Re\times R(1-e)$. My ...
3
votes
0answers
48 views

Degree of the minimal polynomial of the sum of two integral elements over a UFD

Let $D$ be an integral domain ($D$ is a noetherian UFD, if necessary) and let $a,b$ integral over $D$. Let $f$ be the minimal polynomial of $a$ over $D$ and assume it is of degree $n>1$, and let ...
1
vote
2answers
45 views

Can I say that $R= Rr + I$?

Let $I$ be a maximal left ideal of a ring $R$. I have $y \in radR$ e $r \in R$. I am assuming that $yr \notin I$. Can I say that $R= Rr + I$? Definition: $rad R$ is a intesection of the maximal left ...
1
vote
1answer
27 views

Factorization of polynomial in a ring

I want to show that $x^2-14$ doesn't factorize into $(ax+b)(cx+d)$ in $Z_{2014}$ Since $ac=1$ , $x=c^{-1}(-b)$ or $x=a^{-1}(-d)$ is one of the solution "$f(x)=0$" where $f(x)=x^2-14$ and those are in ...
0
votes
2answers
35 views

Where does the proof of unique factorization fail for $\mathbb Z[\sqrt{-5}]$?

I know that unique factorization does not hold for all rings, such has the much-used example $\mathbb Z[\sqrt{-5}]$. It seems that Euclid's lemma does not hold for these rings, and so on. However, ...
2
votes
1answer
47 views

Show that $(\mathbb{Z}/n)^\times$ cannot be cyclic unless $n$ is either a prime power or twice a prime power.

Q: Show that $(\mathbb{Z}/n)^\times$ cannot be cyclic (i.e., there is no primitive root modulo $n$) unless $n$ is either a prime power or twice a prime power. I'm reading a solution. The first ...