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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3
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1answer
86 views

Given a balanced bimodule $_SP_R$, is $R$ isomorphic to $\text{Hom}_S(P,P)$?

For clarity's sake, let me recall some definitions: Given two rings $R$ and $S$, we call a bimodule $_SP_R$ balanced if the ring homomorphisms $$\lambda_P:S \rightarrow \text{End}(P_R)$$ $$\rho_P:R ...
4
votes
2answers
470 views

Principal ideal and free module

Let $R$ be a commutative ring and $I$ be an ideal of $R$. Is it true that $I$ is a principal ideal if and only if $I$ is a free $R$-module?
6
votes
2answers
469 views

Show that a ring with disconnected spectrum is a product of two subrings. [duplicate]

It's an exercise from the book introduction to commutative algebra by Atiyah and Macdonald. If $\operatorname{Spec}(A)$ is disconnected, I'm asked to show that $A$ is a product of two subrings. I ...
4
votes
1answer
90 views

Symmetric and exterior powers of a projective (flat) module are projective (flat)

Assume that $R$ is a commutative ring with unity and $P$ a projective (flat) $R$-module. Why $\mathrm{Sym}^n(P)$ and $\Lambda^n(P)$ are projective (flat) for every $n$?
1
vote
1answer
127 views

Find the multiplicative inverse of x in a quotient ring.

This isn't exactly homework but I'm reviewing past papers (without solutions for an exam), and need some help with the following question: Find the multiplicative inverse of x in the quotient ring ...
4
votes
1answer
138 views

the field $\mathbb{Z}_3[i]$ is ring-isomorphic to the field $\mathbb{Z}_3[x]/(x^2 + 1)$

Let $\mathbb{Z}_3[i] =${$a+bi | a, b \in \mathbb{Z}_3$} . Show that the field $\mathbb{Z}_3[i]$ is ring-isomorphic to the field $\mathbb{Z}_3[x]/(x^2 + 1)$ how can I able to do this?can someone ...
1
vote
1answer
228 views

Why is the additive identity of a ring always a multiplicative absorbing element?

In problems concerned with finding the units in a ring, my textbook seems to always ignore the additive identity as a possibility. In combination with learning the definition of a field (a ring in ...
4
votes
1answer
58 views

$F[x]/\langle p(x)\rangle$ is a field $\iff F[x]/\langle p(x)\rangle$ is an integral domain

$$\color{red}{Is~my~interpretation~correct?}$$ Let $F$ be a field. I know that $p(x)\in F[x]$ is irreducible $\iff \langle p(x)\rangle$ is maximal i.e. $F[x]/\langle p(x)\rangle$ is a field $\iff ...
1
vote
1answer
77 views

Two problems in ideal and radical

Let $R$ be a commutative ring with multiplicative identity. Let $I$ be an ideal of $R$. Let $S=\{r \in R: r^n \in I\mbox{ for some natural number }n\}$. Show that $S$ is an ideal of $R$. Give an ...
3
votes
0answers
115 views

associated graded ring is the quotient of a free algebra by a homogeneous ideal

Let $A$ be a semilocal ring with Jacobson radical $m$ and let $I$ be an ideal of definition, i.e. an ideal such that $m^{\nu} \subset I \subset m$. Consider the associated graded ring of $A$, given by ...
3
votes
1answer
66 views

Ring structure on subsets of the natural numbers

Let $$\mathcal{N}=\{\{k_1,\ldots,k_s\}:\ s>0,\ \mbox{and the}\ k_i\ \mbox{are non-negative and pairwise different integers}\}\cup\{\emptyset\}.$$ Note that there is a bijection with the naturals, ...
1
vote
1answer
93 views

maximal ideal properly contains union of its square with the union of minimal prime ideals

One of the first theorems one encounters in the study of commutative algebra is that if $I$ is an ideal of a ring $A$ not contained in any of the prime ideals $P_1,\cdots,P_n$, then $I$ is not ...
2
votes
1answer
224 views

Why ring with only even numbers is not an integral domain?

Let $S$ be a set of all even integers. According to my text book, $(S,+,\cdot)$ is a ring which is not an integral domain. It is stated as a fact without an explanation and I fail to see the reason ...
1
vote
1answer
36 views

Let $I$ be an ideal in $R$. Show that $\mathrm{ann}_{R}(R/I)=I$.

I am trying to show that $\mathrm{ann}_{R}(R/I)=I$, but not sure whether I am "cheating". I can do the inclusion $I\subseteq \mathrm{ann}_{R}(R/I)$. What I am concerned about is ...
4
votes
1answer
99 views

A question on factorial rings

Is 31 irreducible in the ring $\mathbb{Z}\left[\sqrt{5}\right]=\left\{a+b\sqrt{5}:a,b\in\mathbb{Z}\right\}$ ? And is it prime in $\mathbb{Z}\left[\sqrt{5}\right]$?
3
votes
2answers
153 views

Endomorphisms of a semisimple module

Is there an easy way to see the following: Given a $k$-algebra $A$, with $k$ a field, and a finite dimensional semisimple $A$-module $M$. Look at the natural map $A \to \mathrm{End}_k(M)$ that sends ...
3
votes
2answers
100 views

Proving that tensor distributes over biproduct in an additive monoidal category

I'm trying to prove that the tensor product distributes over biproducts in an additive monoidal category; namely that given objects $A,B,C$, we have $A \otimes (B \oplus C) \cong (A \otimes B) \oplus ...
0
votes
1answer
95 views

Are these prime ideals?

Let $R=\mathbb Z[\sqrt{-5}]$. I want to show $P=3\,R+(1+\sqrt{-5})\,R$ and $Q= 3\,R+(1-\sqrt{-5})\,R$ are prime ideals of $R$.
1
vote
1answer
54 views

$R=\mathbb{Q}[x]/I$ where $I=\langle 1+x^2 \rangle$ let $y$ be the coset of $x$ in $R$ then

$R=\mathbb{Q}[x]/I$ where $I=\langle 1+x^2 \rangle$ let $y$ be the coset of $x$ in $R$ then $y^2+1$ is irreducible over $R$ $y^2-y+1$ is irreducible over $R$ $y^2+y+1$ is irreducible over $R$ ...
2
votes
1answer
83 views

Completion of rings

Theorem 10.1, p.52 of Lang's book on Algebra proves that $\hat{R}_I=\varprojlim_n R/I^n$ is ring ismorphic to Cauchy sequences modulo null sequences, he calls the latter the completion with respect to ...
1
vote
2answers
313 views

Which ring homomorphisms preserve/reflect what?

Exams are coming up and I'm getting kind of desperate. So more now than ever, whatever help you're able to provide is much appreciated. In the abstract algebra exam I'm currently preparing for, ...
5
votes
1answer
141 views

A simpler proof that if $m\mid n$ then there is a ring homomorphism from $\mathbb{Z}_n$ onto $\mathbb{Z}_m$?

Question 18.I.4 from Pinter's A Book of Abstract Algebra asks for a proof of the following, where $\mathbb{Z}_m$ and $\mathbb{Z}_n$ are treated as rings: If $n$ is a multiple of $m$, then ...
3
votes
3answers
358 views

How to prove that $Z[i]/7$ is a field?

I know that $Q[i]$ is a field since we can find the inverse for each $a+bi$ namely $\frac{a}{a^2+b^2} - i\frac{b}{a^2+b^2}$. However, I am not sure how to do so for something like $Z[i]/7$ since we ...
3
votes
5answers
165 views

Rings of fractions and homomorphisms

Let the ring of fractions be denoted as $S^{-1}R$. Proposition (from Robert Ash's textbook "Basic Abstract Algebra") Define $f: R \rightarrow S^{-1}R$ by $f(a) = a/1$. Then f is a ring ...
2
votes
1answer
99 views

Consequence of the Chinese Remainder Theorem

We want to prove the following: For any $n+1$ distinct real numbers $a_0, a_1, ..., a_n$ and any $n+1$ real numbers $b_0, b_1, ..., b_n$, there exists a polynomial of degree at most $n$ taking the ...
11
votes
1answer
238 views

Must an ideal contain the kernel for its image to be an ideal?

I'm trying to learn some basic abstract algebra from Pinter's A Book of Abstract Algebra and I find myself puzzled by the following simple question about ring homomorphisms: Let $A$ and $B$ be ...
10
votes
1answer
333 views

Isomorphic rings or not?

Prove that the ring $\mathbb F_2(T^2)[X]/((X^2+T^2)^2)$ is (or is not) isomorphic to $\mathbb F_2(T)[Y]/(Y^2)$. Remark. The above question is related to this topic, where it's proved that for ...
0
votes
3answers
385 views

Which of these Rings are Euclidean Domains and which are UFDs?

I want to determine which of the following cases are the rings Euclidean Domains and in which they are UFDs. $\mathbb{Q}[X]$ $\bigcup_{n=1}^{\infty}\mathbb{Q}[x^\frac{1}{n}]$ $\mathbb{Q}[X,Y,Z]$ ...
2
votes
1answer
147 views

Extended ideals in power series ring

Let $A$ be a commutative ring with $1$ and consider the ring of formal power series $A[[X]]$. If $I \subseteq A$ is an ideal, let $I[[X]]$ denote the set of power series with coefficients in $I$. This ...
0
votes
2answers
310 views

Every finitely generated algebra over a field is a Jacobson ring

Knowing that the polynomial ring in $n$ variables over a field $k$ is a Jacobson ring, how can we prove from it that every finitely generated $k$-algebra is a Jacobson ring? EDIT: We define a ...
2
votes
2answers
275 views

A question about the relationship between submodule and ideal

It is stated in Wikipedia that if $I$ is an ideal of $R$, that is $I\triangleleft R$, then $_R I$ is a submodule of $_R R$. Here, I assume $R$ is commutative. Despite the notation, I mean ideal is ...
1
vote
1answer
77 views

Why do Artinian rings have dimension 0 and not 1?

One of the properties of an Artinian ring $R$ is that every prime ideal is maximal. So, if $\mathfrak{m}$ is a nonzero prime ideal, $(0)\subseteq \mathfrak{m}$ is a length-$1$ chain of prime ideals, ...
2
votes
0answers
70 views

A principal ideal domain that is not a Euclidean domain. [duplicate]

Somebody can to help me in the following problem: Let R be the following subring of the complex numbers: $$R = \left\{\frac{z_1}{2}+\frac{z_2\sqrt{-19}}{2} : z_1,z_2\in\mathbb{Z}, \;\text{with the ...
2
votes
1answer
233 views

Show that there are irreducible polynomials of every degree in $\mathbb{Q}[X]$

There is this problem that I would like to ask for any verification whether my answer is correct. Edited: Thanks @andybenji. Show that for any $n\ge1$, there exists an irreducible polynomial ...
5
votes
4answers
374 views

Why is $X^4-16X^2+4$ irreducible in $\mathbb{Q}[X]$?

Determine whether $X^4-16X^2+4$ is irreducible in $\mathbb{Q}[X]$. To solve this problem, I reasoned that since $X^4-16X^2+4$ has no rational roots hence irreducible. But there is a hint to this ...
0
votes
0answers
62 views

Generalized division algorithm

Motivated by this question: What is the kernel of the evaluation homomorphism? I would like to know where I can find more about such theorem in $R[x_1,\ldots,x_n]$ instead of in $R[x]$. Thanks in ...
1
vote
1answer
142 views

Quotient of a polynomial ring by a polynomial is equal to the direct sum of quotients by the roots

Reading through Claudio Procesi's Lie Groups: An Approach through Invariants and Representations, I came across the following claim, stated without proof during the derivation of some properties of ...
2
votes
4answers
65 views

Express $18X^2-12X+48$ in $\mathbb{Q}[X]$ as the product of a constant polynomial and a primitive polynomial

I believe I get the correct answer for the following problem but the solution says it is wrong. I have no idea why. Express $18X^2-12X+48$ in $\mathbb{Z}[X]$ and in $\mathbb{Q}[X]$ as the product ...
2
votes
1answer
92 views

Show that “In a UFD, if $p$ is irreducible and $p\mid a$, then $p$ appears in every factorization of $a$” is false

This statement below is false, but I cannot find any counterexamples or explain why. When I tried to give some reasoning, I ended up showing the statement is true. In a UFD, if $p$ is irreducible ...
2
votes
1answer
70 views

Multiplication of rings is an abelian group homomorphism

Let $R$ be a ring without identity. Suppose that the multiplication $ \cdot : R \times R \rightarrow R $ is an abelian group homomorphism. For $a, b \in R$ what can we conclude about the product ...
3
votes
1answer
84 views

Does this complex remain exact after I restrict the maps?

$R$ is a commutative ring with unity. Assume you have two matrices $A:R^n\rightarrow R^m$ and $B:R^m\rightarrow R^n$ such that they form an exact complex in the obvious way, i.e., $$\cdots\rightarrow ...
2
votes
0answers
65 views

Functors that have a natural Isomorphism

Find different functors $T, S: Rng \rightarrow Rng$ both identity on objects IE: for each ring $R, T(R)=S(R)=R$, such that there is a natural isomorphism between T and S. I know that a natural ...
4
votes
2answers
219 views

Is every field the field of fractions for some integral domain?

Given an integral domain $R$, one can construct its field of fractions (or quotients) $\operatorname{Quot}(R)$ which is of course a field. Does every field arise in this way? That is: Given a ...
0
votes
3answers
96 views

Some questions regarding maximal ideals in $\mathbb{Z}$

It is said that the integers is a local ring and hence has a unique maximal ideal. I have two doubts: Why $(p)$ with $p$ prime is a maximal ideal? how can we know it is not possible to have ...
4
votes
1answer
52 views

The identifications of $R$ in its ring of fractions $S^{-1}R$

If $R$ is an integral domain, we can identify the elements $r\in R$ as elements $rs/s$ of the ring of fractions $S^{-1}R$. In this way, we can identify $r\in R$ as $r/1_R$. I've seen in somewhere that ...
2
votes
2answers
53 views

Why $\deg(f)\ge1$ is required in the hypothesis of this statement?

There is this statement that I don't quite understand its hypothesis. Let $R$ be a UFD and $F$ its field of quotients. If $f\in R[X]$ is irreducible in $R[X]$ and $\deg(f)\ge1$, then $f$ is ...
31
votes
4answers
819 views

$\mathbb C[X]/(X^2)$ is isomorphic to $\mathbb R[Y]/((Y^2+1)^2)$

This question led me to the following: Prove that $\mathbb C[X]/(X^2)$ is isomorphic to $\mathbb R[Y]/((Y^2+1)^2)$.
2
votes
1answer
414 views

Ring without zero divisors that has positive characteristic must have prime characteristic

Let $R$ be an integral domain and suppose $R$ has characteristic $n > 0$. Prove that $n$ must be prime. I just proved this exercise, but I think it needs extra conditions. We can prove the ...
5
votes
6answers
225 views

Why does $\mathbb{Z}$ satisfy the ACC?

This might be a really trivial question but somehow my reasoning shows the opposite of what it is supposed to be. Show that the set of integers $\mathbb{Z}$ satisfies the Ascending Chain ...
2
votes
1answer
359 views

Determine the maximal ideals of the rings $\mathbb{C}[X]$, $\mathbb{R}[X]/(X^2)$, $\mathbb{R}[X]/(X^2+1)$, $\mathbb{C}[X]/(X^2+1)$

I am trying to determine the maximal ideals in the following rings: 1. $\mathbb{C}[X]$ 2. $\mathbb{R}[X]/(X^2)$ 3. $\mathbb{R}[X]/(X^2+1)$ 4. $\mathbb{C}[X]/(X^2+1)$ By reasoning as follows: An ...