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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1
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2answers
96 views

Nilpotent or non-Nilpotent Jacobson Radical

Let $R$ be a ring with identity element such that every ideal of which is idempotent or nilpotent. Is it true that the Jacobson radical $J(R)$ of $R$ is nilpotent? If $R$ is Noetherian and $J(R)$ is ...
14
votes
1answer
77 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 ...
5
votes
1answer
1k views

Correspondence theorem for rings.

Could someone provide a reference that includes a full and honest proof of the Correspondence Theorem for rings? Let $A$ be a multiplicative ring with identity and $I$ an ideal of $A$. There is a ...
0
votes
3answers
24 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 ...
3
votes
3answers
501 views

Showing that the elements $6$ and $2+2\sqrt{5}$ in $\mathbb{Z}[\sqrt{5}]$ have no gcd

In showing that the elements $6$ and $2+2\sqrt{5}$ in $\mathbb{Z}[\sqrt{5}]$ have no gcd, I was thinking of trying the following method. If the ideal $(6)$ + $(2+2\sqrt{5})$ is not principal in $\...
3
votes
0answers
33 views

$R$ 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
32 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
2answers
43 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 $\langle q(x) \rangle$ ...
2
votes
1answer
39 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, ...
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
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1answer
23 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
37 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
2answers
41 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 ...
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 $\...
1
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2answers
49 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 ...
0
votes
2answers
30 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
28 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 ...
2
votes
1answer
30 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 ...
4
votes
1answer
47 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
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?
11
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1answer
168 views
+100

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 ...
3
votes
1answer
92 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
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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 ...
2
votes
1answer
30 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
0answers
46 views
+100

Does every finite dimensional real nil algebra admit a multiplicative basis?

We say that a finite dimensional real commutative and associative algebra $\mathcal{A}$ is nil if every element $a \in \mathcal{A}$ is nilpotent. By multiplicative basis, I mean a basis $\{ v_1, \...
1
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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.
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
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1answer
79 views

Does reduceness of $K[t_1,\dots,t_n]/I$ imply radicality of $I$?

Let $I$ be an ideal in $K[t_1,\dots,t_n]$. Is it true that if the quotient $K[t_1,\dots,t_n]/I$ is reduced then $I$ is radical? We say that a ring $R$ is reduced if $x^2 = 0$ implies $x=0$ for all $x ...
6
votes
1answer
23 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 ...
1
vote
1answer
22 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
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-...
0
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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
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 ...
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0answers
20 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
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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 ...
2
votes
2answers
59 views

Showing $\mathbb Z[x]/<5,x^3+x+1>$ is a field

I want to show that $ \mathbb Z[x] /<5,x^3+x+1>$ I am currently self-studying and it is a problem of previous graduate school entrance exam. I studied abstract algebra through Fraleigh's book, ...
15
votes
4answers
3k views

Zero divisor in $R[x]$

Let $R$ be commutative ring with no (nonzero) nilpotents. If $f(x) = a_0+a_1x+\cdots+a_nx^n$ is a zero divisor in $R[x]$, how do I show there's an element $b \ne 0$ in $R$ such that $ba_0=ba_1=\cdots=...
-1
votes
0answers
45 views

a simple problem about commutative algebra [duplicate]

I think this problem seem easy, but I have no idea to approach this. Let $R$ is commutative Ring, a non-zero $f\in R[X]$ is zerodivisor if and only if there exists no-zero $c\in R$ such that $c.f=0$...
1
vote
1answer
26 views

Extensions between integral domains give extensions of fields of the same degree.

Assume that $S \subset R$ is a ring extension where, both $S$, $R$ are integral domains. Furthermore, assume that $R$ is a free $S$-module of rank $n$. Is it true that the extension of fields $\mathrm{...
0
votes
1answer
30 views

What does “coefficients from all of $\mathbf{F} _q$” mean

I was reading Wikipedia's page on Ring Learning with Errors, and came to wonder what is meant by "with coefficients from all of $\mathbf{F} _q$" which is a requirement for the set of known polynomials....
2
votes
1answer
34 views

Mal'cev condition for variety of rings generated by finite fields to be arithmetical

This is an exercise of Burris & Sankappanavar (Universal Algebra), Chapter II, section 12. It asks to prove that, if $V$ is a variety of rings generated by finitely many finite fields, then $V$ is ...
4
votes
2answers
347 views

Two questions about integral “splitting ring” extensions

We have a ring $R$, commutative, and $f_1,\dots,f_n$ polynomials in $R[x]$ monic, with $\deg f_i\ge 1$. It is straightforward to show that there is a ring extension $R\subset S$ such that $S$ contains ...
1
vote
4answers
65 views

Showing that two quotient rings are isomorphic

Is $\mathbb{Q}[x]/(x^2-x-1)$ isomorphic to $\mathbb{Q}[x]/(x^2-5)?$ My guess is yes. I am trying to find an isomorphism between the two. Universal Property of Quotient certainly helps. I am thinking ...