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 (2)

1
vote
1answer
14 views

$R$ be an infinite commutative ring such that $R/I$ has only finitely many ideals for every non-zero ideal $I$ , what can we say about $R$?

It is known that if $R$ is an infinite commutative ring such that for every non-zero ideal $I$ , $R/I$ is finite then $R$ is a Noetheian domain . It is also known that if $R$ is a PID then for every ...
1
vote
1answer
26 views

Matrices representing injective homomorphisms

Let $R$ be a ring and $M$, $N$ finitely generated free modules modules over $R$. Let $A$ be a matrix representing a homomorphism $f: M \rightarrow N$. We know that the map $f$ is injective if and only ...
0
votes
0answers
26 views

Elements of $\mathbb Z [i]/\langle 1+4i\rangle$

What are the elements of $\mathbb Z [i]/\langle 1+4i\rangle$ ? I know that there are 17 elements as the norm of $1+4i$ is 17 but I can't manage to find them...
4
votes
2answers
41 views

Two questions on the Gaussian integers [duplicate]

I have two questions on the Gaussian integers. Is any element in $\mathbb{Z}[i]$ the root of a monic polynomial with coefficients in $\mathbb{Z}$? Conversely, does any element in $\mathbb{Q}(i)$ ...
5
votes
2answers
734 views

Irreducible but not prime element

I am looking for a ring element which is irreducible but not prime. So necessarily the ring can't be a PID. My idea was to consider $R=K[x,y]$ and $x+y\in R$. This is irreducible because in any ...
0
votes
1answer
46 views

Does there exist any non-zero polynomial $f:\mathbb C \to \mathbb C$ such that $f(x+2)-2f(x+1)=f(x) , \forall x \in \mathbb C$ ?

Does there exist any non-zero polynomial $f:\mathbb C \to \mathbb C$ such that $f(x+2)-2f(x+1)=f(x) , \forall x \in \mathbb C$ ?
2
votes
1answer
44 views

Is $4x^2+3xy^2+y^3+7$ irreducible in $\mathbb{C}[x,y]$?

Is $4x^2+3xy^2+y^3+7$ irreducible in $\mathbb{C}[x,y]$? I tried to group it like $(4)x^2+(3y^2)x+(y^3+7)$. This is a polynomial with degree $2$ so I am thinking of applying quadratic formula... where ...
-4
votes
1answer
58 views

Set of zero divisors is an ideal iff the ring is local [on hold]

Let $R$ be a commutative ring with unity. Show that $Z(R)$, the set of all zero divisors of $R$, is an ideal if and only if $R$ is a local ring. I have no idea for proving this. Thanks in advance!
7
votes
0answers
56 views

If $A[X] \cong B[X]$ as rings, are the degrees of irreducible polynomials the same in $A$ and in $B$?

First, I ask my question and then I add some explanations: Suppose that $A$ and $B$ are two commutative rings such that $A[X] \cong B[X]$ as rings. Denote by $D_A$ the set of all positive integers ...
3
votes
1answer
53 views

Is there a theory of generalized eigenvectors over commutative rings?

Brown's Matrices over Commutative Rings book discusses the theory of eigenvalues, eigenvectors, and diagonalizing matrices over commutative rings, but unless I've missed something, nothing like ...
2
votes
1answer
41 views

Norm of Gaussian integers: solutions to $N(a) = k$ for $k \in \mathbb{N}$ and $a$ a Guassian integer

I have two questions. Which integers are equal to the norm of some Gaussian integer? In general, how many solutions does$$\text{N}(a) = k$$have for a given $k \in \mathbb{Z}$? I am investigating the ...
4
votes
2answers
62 views

$R$ be a commutative unital ring , is it true that the group of units of $R$ is not isomorphic with the additive group of $R$?

Let $R$ be a commutative ring with unity , let $R^{\times}$ be the group of units of $R$ , then is it true that $(R,+)$ and $(R^{\times} ,\cdot)$ are not isomorphic as groups ? I know that the ...
0
votes
1answer
27 views

A question on extension of rings which related to their direct summands

I read "Foundations of Module and Ring Theory" of Robert Wisbauer and I got stuck in this problem: *Show for a ring $R$. The following assertions are equivalent: (a) $R$ has a unit. (b) If $R$ is ...
16
votes
1answer
178 views

Does $A^2 \cong B^2$ imply $A \cong B$ for rings?

If $A$ and $B$ are two unital rings such that $A \times A \cong B \times B$, as rings, does it follows that $A$ and $B$ are isomorphic (as rings)? I believe that the answer is no, but I can't come ...
0
votes
3answers
25 views

Minimal ideal in a ring which is generated by an idempotent element.

Let $R$ be a commutative ring with unity and $M$ be a minimal ideal of $R$ such that $M = Re$ where $e$ is an idempotent element in $R$. Then $R = Re \oplus R(1-e) $ I am not able to see, in order ...
4
votes
1answer
58 views
+50

$R$ be an integral domain , $x \in R$ , $I$ an ideal such that $I+\langle x \rangle , (I:x)$ are principal ideals , then is $I$ a principal ideal?

Let $R$ be an integral domain , $x \in R$ , $I$ be an ideal such that $I+\langle x \rangle $ and $(I:x):=\{r \in R : rx \in I\}$ both are principal ideals , then is $I$ also a principal ideal ?
0
votes
0answers
35 views

Algebra of Endomorphisms of a functor $\mathcal{F}$

Recently, I was reading a paper and stumbled upon something for which I don't find the formal definition unfortunately in any textbook that have in my hands. It's the notion of the endomorphism ...
2
votes
1answer
43 views

$\Bbb Z[\sqrt{-5}]$ is not a PID [duplicate]

I want to show: In a PID $R$ two elements $a,b\in R$ always have a greatest common divisor. Therefore $\Bbb Z[\sqrt{-5}]$ is not a PID. For the first part: $I=\{ax+by:x,y\in R\}$ is an ideal, ...
2
votes
2answers
45 views

What are the conditions needed for two principal ideals of a ring to be isomorphic?

Given a commutative ring $R$, and $p(x),q(x) \in R[x]$ monic polynomials, under what conditions on $p(x)$ and $q(x)$ are the principal ideals $\langle p(x) \rangle$ and $\langle q(x) \rangle$ ...
0
votes
1answer
29 views

Is $\Bbb Z[i]$ a Euclidean ring? [duplicate]

Is $\Bbb Z[i]$ a Euclidean ring? If not, what would be the simplest way of seeing that $\Bbb Z[i]$ is a PID?
1
vote
1answer
24 views

Given a commutative ring $R$ and a monic polynomial $p(x) \in R[x]$ is $R[x]/\langle p(x) \rangle$ always a finite integral extension of $R$?

I suspect this to be true based on the fact that $p(x)$ is monic, so it should be the case that $R[x]/\langle p(x) \rangle$ is a finitely generated module over $R$, but I have no good reference for ...
4
votes
1answer
50 views

Quotient ring of Gaussian integers $\mathbb{Z}[i]/(a+bi)$ when $a$ and $b$ are NOT coprime

The isomorphism $\mathbb{Z}[i]/(a+bi) \cong \Bbb Z/(a^2+b^2)\Bbb Z$ is well-known, when the integers $a$ and $b$ are coprime. But what happens when they are not coprime, say $(a,b)=d>1$? — For ...
1
vote
1answer
34 views

Proving that a prime ideal $p \subset R$ yields a prime ideal $p[x] \subset R[x]$

I'm curious as to whether I can have my proof critiqued. Proposition : Let $\mathfrak{p}$ be a prime ideal in a ring $R$. Show that $\mathfrak{p}[x]$ is a prime ideal in $R[x]$. Proof : Suppose $\...
0
votes
2answers
32 views

$R/Rg$ is a field iff $g\in R$ is irreducible.

Let $R$ be a PID and $g\in R$. I want to show: $R/Rg$ is a field iff $g\in R$ is irreducible. I.e. I want to show that all $a\notin Rg$ are invertible modulo $g$ iff $g$ is irreducible. So if I ...
2
votes
1answer
29 views

Bijection beteween maximal ideals

We know that if $R$ and $I$ an ideal of $R$, then there is a bijection between the prime ideals of $R$ containing $I$ and the prime ideals of $R/I$. It is given by $P\mapsto P/I$. Is it true that this ...
1
vote
2answers
50 views

Prove that if $I$ is maximal, then $R[X]$ is a PID. [duplicate]

Let $R$ be a commutative ring with unity such that $R[X]$ is a UFD. Denote the ideal $\langle X\rangle $ by $I$. Prove that If $I$ is maximal, then $R[X]$ is a PID. If $R[X]$ is a Euclidean Domain ...
4
votes
1answer
48 views

Prove that up-to isomorphism there are two integral domains of order $p^2$.

Prove that up-to isomorphism there is exactly one integral domain of order $p^2$ . Does there exist only two non-commutative rings of order $p^2$ upto isomorphism? We know that any group of ...
3
votes
1answer
95 views

Is it possible to endow $\text{GL}_2(\Bbb R)$ with a ring structure?

My question is the following: Is it possible to find a binary operation $*$, seen as an addition, such that $(\text{GL}_2(\Bbb R),*,\cdot)$ has a ring structure (not necessarily with a unit)? [We ...
2
votes
0answers
17 views

Example for Jacobson density theorem

I'm reading through Lang's algebra. Lang gives the Jacobson density theorem in the following way: Let $R$ be a ring (with unity) and $E$ a semisimple $R$-module. Let $R' = \operatorname{End}_R(E)...
1
vote
2answers
47 views

Definition of a simple ring

I'm reading through Lang's Algebra. Lang defines a simple ring as a semisimple ring that has only one isomorphism class of simple left ideals. On the other side, Wikipedia says that a simple ring is ...
1
vote
2answers
43 views

Left- and right-sided principal ideals have same index?

One fact about the Lipschitz integers (quaternions of the form $a + bi + cj + dk$ where $a, b, c, d$ are integers) is that the left-sided ideal generated by any element $Q$ has the same index in the ...
2
votes
1answer
32 views

Why is the Rees Algebra Noetherian if the underlying ring is?

Let $R$ be a commutative ring with $1$, $I \subset R$ a proper ideal. The Rees Algebra, with respect to $I$, is defined: $R[It]= \bigoplus_{n=0}^\infty I^nt^n \subseteq R[t]$. In many places I've read ...
1
vote
1answer
29 views

Algebra Generated by a set modulo relations

I have a very basic question I think, but it's something that can't find in literature. So, lots of times happen to see in a book a phrase of the form "Thus, we can define the free associative $\...
3
votes
2answers
63 views

Is a finite inverse limit of noetherian rings noetherian?

Let $\{A_i\}$ be an inverse system of (commutative, unital) Noetherian rings with a finite index set. Is $\varprojlim A_i$ also a Noetherian ring?
1
vote
1answer
28 views

Ring isomorphism $\Bbb Q[x]/(f)\cong \{c_0+c_1\alpha + c_2\alpha^2:c_i\in \Bbb R\}$

Let $f=x^3+x^2-2x-1\in \Bbb Q[x]$. Let $\alpha\in \Bbb R$ be a zero of $f$. $\Bbb Q[x]/(f)$ is isomorphic to the subring $R=\{c_0+c_1\alpha + c_2\alpha^2:c_i\in \Bbb R\}$ of $\Bbb R$. The map $\...
1
vote
2answers
29 views

Extension of a finite field to a finite non commutative ring

Can a finite field be extended to non-commutative finite rings so that not all elements of the field commutes with the elements of the ring? I have been trying this taking the examples of matrices.
6
votes
1answer
24 views

Relation between semiring of sets and semiring in abstract algebra.

Let a $\mathcal R$ be a family of subsets in $\Omega$ that is closed under finite union and relative complement. We say that $\mathcal R$ is a ring of sets in $\Omega$. Symbolically, for any $A,B\in\...
0
votes
1answer
18 views

$\mathbb{F}$-subalgebra generated by a set

Assume that $A$ is an $\mathbb{F}$-algebra, where by $\mathbb{F}$ I just denote an arbitrary field. Furthermore, if $X \subset A$ is a proper subset, how do we define the $\mathbb{F}$-subalgebra of $A$...
0
votes
0answers
24 views

Trivial extension of an opposite algebra

Suppose that $A$ is a finite dimensional $K$-algebra, where $K$ denote an algebraically closed field. Call $DA=Hom_k(A,k)$. $DA$ admits an $A$-$A$-bimodule structure in the obvious way. The trivial ...
2
votes
1answer
34 views
+50

Upper bound for the no. of maximal ideals of a finite commutative ring ( with unity ) in terms of the order of the ring and/or its number of units?

Is there any known upper bound for the no. of maximal ideals of a finite commutative ring ( with unity ) of order $n$ , in terms of $n$ and /or in terms of the no. of units of the ring ? Say , does ...
1
vote
1answer
23 views

Annihilator of maximal ideals

Let $R$ be a Noetherian ring. Suppose all the nonzero proper ideals of $R$ have nonzero annihilators. Show that if $M$ is a maximal ideal of $R$ , then $\exists$ $x \in R $ such that $M$ = $ann(x)$ (...
2
votes
0answers
58 views

Prime ideal of a polynomial ring in 6 variables

Let $k$ be a field and $k[x_1,x_2,x_3,y_1,y_2,y_3]$ a polynomial ring in 6 variables over $k$. How to prove that the ideal $(x_1y_2-x_2y_1,x_2y_3-x_3y_2,x_3y_1-x_1y_3)$ is prime in $k[x_1,x_2,x_3,y_1,...
0
votes
0answers
21 views

Rings and modules-ideals and submodules

I am taking a course in Commutative Algebra and the following lemma in a section on localisation raised some questions. Lemma: Let M be an R-module. The following are equivalent. (1) $M=0$ (2) $M_P=...
0
votes
0answers
27 views

Questions Concerning Proof of Artin-Rees Lemma

I have two questions about the proof of the Artin-Rees Lemma presented here: $\textit{Question 1:}$ Am I correct in assuming $I^0=R$? This is the only way some of the statements make sense I think. ...
0
votes
3answers
67 views

An elementary proof that $k[x,y]/(xy-1)\cong k[x]_x$, where $k$ is a field

Letting $\phi:k[x,y]\to k[x]_x$, $\phi(x)=x$, $\phi(y)=\frac{1}{x}$, we see that $\ker \phi$ is prime, and $(1-xy)\subseteq\ker\phi$. Now, given that $k[x,y]$ has Krull dimension 2, $\ker\phi\neq (1-...
11
votes
1answer
189 views

Why does this ring have rank $k!$?

Let $R$ be any ring free of rank $k$ over $\mathbb{Z}$ having non-zero discriminant. Let $R^{\otimes k} = R \otimes_{\mathbb{Z}} \cdots \otimes_{\mathbb{Z}} R$. Then $R^{\otimes k}$ is a ring of rank ...
0
votes
0answers
39 views

Advantages and disadvantages of a particular definition of rings and subrings

My professor defines a ring as a triple $(A, +,\ \cdot)$ such that $(A, +, 0)$ is an abelian group, $(A,\ \cdot)$ is a semigroup and $\cdot$ distributes over $+$. Subsequently, $B\subseteq A$ is said ...
1
vote
0answers
21 views

$f:A\to B$ flat homomorphism of rings, $Q$ is a prime ideal of $B,$ and $P=Q^c.$ Why is $B_Q$ a local ring of $B_P?$

I have a question about Exercise 2.18 on Introduction to Commutative Algebra by Atiyah and MacDonald. Let $f:A\to B$ be a flat homomorphism of rings, let $Q$ be a prime ideal of $B$ and let $P=Q^c,$ ...
0
votes
1answer
13 views

If $B$ is a commutative domain, $Aut(B)$ acts on $Der(B)$ by conjugation

I'm reading Algebraic Theory of Locally Nilpotent Derivations by Gene Freudenberg, and I don't understand what's meant on the line $Aut(B)$ acts on $Der(B)$ by conjugation: $\alpha \cdot D = \alpha ...
1
vote
1answer
29 views

A question about Ring theory

I'm studying basic Ring Theory. And in my textbook, the author states the definition of Euclidean domain: The integral domain $R$ is called to be a Euclidean domain precisely when there is a ...