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)

0
votes
3answers
303 views

$R = \mathbb{Z}[ i ] / (5)$ is not an integral domain? Why?

Let $R = \mathbb{Z}[ i ] / (5)$ . How should I prove that $5 = (2+i) (2-i)$ is a prime factorization in $\mathbb{Z}[i]$? Can we deduce from this that R is not an integral domain? How? I know that ...
6
votes
1answer
251 views

Prove Noether-Skolem theorem for $M_2(\mathbb{C})$ by calculation

Noether-Skolem Theorem for the case the ring is $M_2(\mathbb{C})$ says that "Every $\mathbb{C}$-algebra automorphism of $M_2(\mathbb{C})$ is inner." Now, how to prove it by a direct calculation? ...
2
votes
1answer
85 views

How to prove $A[x]/\langle x\rangle\cong A$ for a ring $A$?

I am working on this question and I was wondering if I was on the right track. It states: "Let $A$ be a ring. Prove that $A[x]/\langle x\rangle$ is isomorphic to $A$. So am I on the right track in ...
5
votes
2answers
159 views

Is there any non-monoid ring which has no maximal ideal?

Is there any non-monoid ring which has no maximal ideal? We know that every commutative ring has at least one maximal ideals -from Advanced Algebra 1 when we are study Modules that makes it as a very ...
0
votes
0answers
192 views

structure of formal linear combinations

Let $S=\{x_1,\cdots,x_k\}$ be a finite set. and let $G$ be an abelian group. let $C(G)$ be a set whose elements are formal linear combinations $\sum{g_ix_i}$ where $g_i \in G$. I want to understand ...
2
votes
3answers
154 views

Software for deciding ideal membership

Let $\alpha$ be such that $\alpha^3 + \alpha + 1 = 0$ and consider $\Bbb{Z}[\alpha]$. Suppose I have an ideal in $\Bbb{Z}[\alpha]$ that is given by $$ I = \Bigg(23^3, 23^2(\alpha - 3), 23(\alpha - ...
7
votes
1answer
240 views

Two questions on Nagata's counterexample to the Hilbert's fourteenth problem.

Let $\{a_{ij}\}$ for $i=1,2,3$, and $j=1,...,16$ be algebraically independent elements over some prime field. Let $k$ be a field containing all $a_{ij}$. Then consider $k^{16}$ as $k$-vector space and ...
12
votes
3answers
531 views

If $x^3 =x$ then $6x=0$ in a ring

Let $R$ be a ring with unity where $$x^3=x,\;\;\; \forall x \in R$$ How do I prove that $$x+x+x+x+x+x=0$$
4
votes
1answer
81 views

What is the definition of 'regular local' and 'regular' for noncommutative rings?

I have been trying to find out what the definition of a noncommutative regular local ring is, but to no avail. In fact, how does one even begin to define Krull dimension for a noncommutative ring? ...
1
vote
1answer
31 views

A $\mathbb Z_2$-gradation on a ring.

Let $R$ be a ring for which we have a $\mathbb Z_2$-gradation meaning that $R=R_0\oplus R_1$, for example when $R=\mathbb C$ we have that $R_0=\mathbb R$ and $R_1=\mathbb R i$. I'm having trouble ...
14
votes
3answers
1k views

Elements in $\hat{\mathbb{Z}}$, the profinite completion of the integers

Let $\hat{\mathbb{Z}}$ be the profinite completion of $\mathbb{Z}$. Since $\hat{\mathbb{Z}}$ is the inverse limit of the rings $\mathbb{Z}/n\mathbb{Z}$, it's a subgroup of $\prod_n ...
0
votes
1answer
51 views

Example of non-unique-factorization in $\mathbb{Z}+x \mathbb{Q}[x]$.

Could you provide me an example of a polynomial that is not uniquely factorized in $\mathbb{Z}+x \mathbb{Q}[x]$? Thanks!
12
votes
5answers
856 views

Finite quotient ring of $\mathbb Z[X]$

Since userxxxxx (I don't remember the numbers) deleted his own question which I find interesting, let me repost it: Let $f,g\in\mathbb Z[X]$ with $\mathrm{gcd}(f,g)=1$. Prove that the ring ...
6
votes
2answers
319 views

Which field is this quotient of a local ring by its maximal ideal?

Let $p\in\mathbb{Z}$ be a prime number, $\mathfrak{p}\subset \mathbb{Z}$ be the prime ideal it generates and let $\mathbb{Z}_{\mathfrak{p}}$ be the localization of $\mathbb{Z}$ at $\mathfrak{p}$, i.e. ...
5
votes
1answer
494 views

Constructing Idempotent Generator of Idempotent Ideal

Exercise 2.1 in Matsumura's Commutative Ring Theory reads as follows: "Let $A$ be a commutative ring and $I$ an ideal that is finitely generated and $I=I^2$. Then $I$ is generated by an idempotent." ...
2
votes
1answer
160 views

Non-Free Finitely Generated Injective Modules over a Local Ring

I was wondering if someone could be so kind as to provide an example of a local ring $ (R,\frak{m}) $ and a non-free finitely generated injective module over $ R $. Thank you very much! I tried ...
3
votes
1answer
192 views

How does one show that this tensor product is not torsion-free?

I am having trouble showing that a particular tensor product is not torsion-free. Let $ R = k[[x,y]] $, where $ k $ is a field (this is the ring of formal power series in $ x $ and $ y $ with ...
3
votes
2answers
285 views

Non-isomorphic rings of given cardinality that are non-commutative

I need to find an example of two non-isomorphic rings of cardinality 16 that are non-commutative. What is the best approach to such problem?
1
vote
0answers
52 views

Is the group ring of a pro-finite group (semi)-hereditary?

It is well-known that a group ring of a finite group is semi-simple, and since profinite-groups are projective limits of finite groups, I am thinking per chance profinite groups still possess some ...
6
votes
1answer
212 views

Is there any deep connection between algebraic topology and homological algebra on rings?

There is a deep connection between algebraic topology and homological algebra on groups. A group $G$ can be interpreted as the fundamental group of a covering space $Y \rightarrow X$. (Co)Homology ...
1
vote
1answer
64 views

Fields arising as endomorphism rings

Do you know a field $K$ other than $F_p$ which is the endomorphism ring of an abelian group $G$? I doubt that there is one because as $G$ gets bigger, $End(G)$ seems to be more and more ...
8
votes
2answers
390 views

Does every ring with unity arise as an endomorphism ring?

I don't believe that every ring with a $1$ is the endomorphism ring of an abelian group but I currently don't see how to produce a counterexample.
9
votes
3answers
964 views

What are applications of rings & groups?

I am following a course in basic algebra, and we have covered rings & groups in class, but I am having trouble visualising them. Are there applications of group &/or ring theory that can be ...
0
votes
1answer
182 views

Variation of the universal property for the field of fractions

Consider the universal property for the fraction field of an integer domain: Let $R$ be a integral domain, $F(R)$, its fraction field, $K$ some field and $f:R\rightarrow K$ a injective ring ...
5
votes
2answers
109 views

How to show $\bigcap_{m \textrm{: maximal ideal}} A_m=A$?

$A$ is an integral domain. For every maximal ideal $m$ in $A$, consider $A_m$ as a subring of the quotient field $K$ of $A$. Show $\bigcap A_m=A$, where the intersection is taken over all maximal ...
0
votes
1answer
213 views

Module Homomorphisms and a Direct Sum

I am trying to show the following: Let $f:M\to N$ and $g:N\to M$ be module homomorphisms such that $g\circ f=Id_M$. Prove that $N=Im(f)\oplus\ker(g)$. I know that $M/\ker(f)\cong Im(f)$, but I'm not ...
0
votes
1answer
59 views

Equivalence of two definitions of an algebra over a ring.

I had previously seen the definition to be the one in Atiyah-Macdonald's Commutative Algebra: A is a ring and an algebra over a ring is a ring B such that there is a map $\phi:A\rightarrow B$. ...
1
vote
3answers
278 views

field of prime characteristic

Say $F$ is a field of characteristic $p$ and let $f(x) = x^p - a \in F[x]$. Show that $f$ is irreducible over $F$ or $f$ splits in $F$. Well, my solution would be since $Char F = p$, then $(x - ...
2
votes
1answer
514 views

Maximal ideals in $\mathbb{R}[x,y]/(xy-2)$?

I'm working on a practice exam and one of the questions asks if there are any maximal ideals in $\mathbb{R}[x,y]/(xy-2)$ and, if so, to find one of them. Initially, I thought the quotient ring was a ...
2
votes
1answer
679 views

An ideal that is radical but not prime.

I'm preparing for an exam and, as part of this preparation, I'm looking for an ideal $I$ in an integral domain $R$ that is radical but not prime. Here is an example I'm fooling around with: Let ...
1
vote
1answer
3k views

How does one prove that a polynomial has no rational roots in general?

How can we prove that a polynomial only has rational roots when we know the coefficients and the degree? For instance, in illustration, how would we show this for $x^8 ...
0
votes
1answer
170 views

Can the Euclidean algorithm prove Euclid's Lemma in a UFD?

Let $A$ be a UFD. Assume that $a,b \in A$ are relatively prime, $c \in A$ and $a | bc$. To prove that $a|c$, is the following approach correct (or do you have to use some type of prime factorization ...
6
votes
1answer
215 views

Questions about subalgebras of finitely generated $k$-algebras

Let $k$ be a field (if necessary assume $k$ to be algebraically closed). Let $A$ be a finitely generated $k$-algebra and let $B$ be a subalgebra of $A$. Remark that $B$ doesn't have to be noetherian, ...
1
vote
1answer
93 views

Abstract Algebra elementary question

Let $F = \mathbb{Q}(\pi^3)$. How can we find a basis for $F(\pi)$ over $F$ ?
4
votes
1answer
106 views

Is the section of a surjective endomorphism again a morphism?

Let $R$ be a commutative ring with unit and let $M$ be an $R$-module (with $1\cdot m = m$ for all $m\in M$). Let $f:M\twoheadrightarrow M$ be a surjective morphism of $R$-modules of $M$ onto itself. ...
1
vote
1answer
133 views

What is Proj $\mathbb{C}[x,y][z]/\langle xz-yz\rangle$?

Assuming that $x,y$ have weight $0$ and $z$ has weight $1$, $$ R= \mathbb{C}[x,y][z]/\langle xz-yz\rangle = \mathbb{C}[x,y]\oplus ( \oplus_{i\geq 1}\mathbb{C}[x]z^i), $$ what closed subvariety is ...
3
votes
2answers
66 views

Some questions of PIDs

I'm stuck on a couple of practice problems relating to PIDs, they are paraphrased below: Given a PID $R$ with $a$ and $b$ in $R$ and gcd$(a,b)=1$ I need to show that: 1) There are elements $s$ and ...
3
votes
2answers
137 views

Self-injective noetherian rings

I would like to prove that the following conditions for a ring $R$ are equivalent: 1) $R$ is Noetherian and self-injective; 2) The class of projective $R$-modules is equal to the class of ...
3
votes
1answer
149 views

Classifying all ideals of a lattice $\mathbb{Z}[\sqrt{-d}]$

In Artin's Algebra he presents a method (that I am sure I am butchering) for classifying ideals of a given lattice $\mathbb{Z}[\sqrt{-d}]$ by taking any ideal $I$, choosing an element of minimum norm ...
0
votes
2answers
215 views

every ideal of $F[x]$ is prime ideal but not maximal

Let $F$ be integer domain but not be field, then every ideal of $F[x]$ is prime ideal but not maximal?. Now, I set a homomorphism $$\varphi: R[x]\to R$$ defined by $\varphi(a_0+a_1 x+\ldots+a_n ...
0
votes
1answer
66 views

When is $f(r)^{-1} \neq f(r^{-1})$

I'm wondering if there are homomorphisms $f$ between unitary rings $R,S$, such that $f(r)$ is invertible but it's inverse doesn't equal $f(r^{-1})$. That is only possible if the inverse isn't in the ...
0
votes
5answers
329 views

Are rings with the same finite cardinality isomorphic?

For fields it is well known, that all fields with finite cardinality $n$ are isomorphic. Does a similar result also hold for rings, i.e. are rings that have the same finite cardinality isomorphic ? ...
1
vote
3answers
219 views

Are $injective$ homomorphisms required for the universal property of the fraction field?

Consider the universal property for the fraction field of an integer domain: Let $R$ be a integral domain, $F(R)$, its fraction field, $K$ some field and $f:R\rightarrow K$ a injective ...
5
votes
1answer
237 views

When is intersection of infinitely many maximal ideals zero?

I've been trying without success to figure out what are the rings $R$ such that whenever $M_n, n \in \omega$ is a countably infinite collection of pairwise distinct maximal ideals then $\bigcap_{n \in ...
0
votes
1answer
83 views

Contrapositive proof of prime ideal of semirings

I'm starting to study about the Ternary A-semirings and studying the paper of Daddi and Pawar entitle: Ideal Theory in Commutative Ternary A-semirings, which was published in Int. Math. Forum vol. 7 ...
3
votes
1answer
65 views

In ring $(R,+,*)$, if $-x\in R$, can we prove (or assume) $x\in R$?

In ring $(R,+,*)$, the minus sign is often given as a unary operator for the additive inverse such that: $\forall x\in R (-x\in R)$ $\forall x\in R(x+(-x)=0 \wedge (-x)+x=0)$ If we have $-x\in R$, ...
8
votes
5answers
1k views

If every prime ideal is maximal, what can we say about the ring?

Suppose $R$ is a ring and every prime ideal of $R$ is also a maximal ideal of $R$. Then what can we say about the ring $R$?
2
votes
2answers
43 views

Ring theory question please?Simple.

We have the Cartesian product $B \times B$ and there we have the addition $$(f,g)+(h,k)=(f+h,g+k)$$ and the multiplication $$(f,g) \cdot (h,k)=(f\cdot h+g\cdot k,f\cdot k+g\cdot h).$$ I want to find ...
2
votes
3answers
106 views

Abelian rings question

I have to prove that the ring $(B,+,*)$ is abelian only when for every $(a,b) \in B^2$, $(a+b)^2=a^2+2ab+b^2$. I don't know where to start, and also I can relate $*$ to the ring, not $+$.
0
votes
1answer
77 views

Can the Natural Numbers (or equivalent) be Constructed from the Ring Axioms?

With only the additional axiom that $0\neq 1$, I think I have been able to formally construct the a subset $n$ of the ring $(R,+,*,0,1)$ using only a subset axiom (specification in ZF). Informally, ...