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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5
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1answer
240 views

Integral dependence and (faithfully) flat ring extensions

Let $R\to S$ be a flat ring extension. By theorem 9.5 of the book Commutative Ring Theory written by Matsumura the going-down theorem holds between $R$ and $S$. Is it true (or not) about these ...
1
vote
2answers
129 views

Specific subring with non-zero unity element

I'm trying to find an example of a ring with unity $1\ne 0$ that has a subring with non-zero unity $1' \ne 1$. Any hints here? Thanks a lot, Mariogs
3
votes
3answers
69 views

Finding indempotents in a quotient ring

I am trying to find the nontrivial indempotents in the ring $\mathbb{Z_3}[x]/(x^2+x+1)$. We can clearly see that $0,1$ are indempotents. I want to prove they are the only ones. Thus I am ...
2
votes
1answer
128 views

Exercise 2.26 Atiyah-Macdonald, flatness

I'm stuck on this exercise. $A$ is a commutative ring with unit. $N$ is an $A$-module. Then $N$ is flat $\Longleftrightarrow $ $\text{Tor}_{1}(A/a, N ) = 0 $ for every finitely generated ideal $a$ of ...
0
votes
1answer
83 views

Direct Product of Rings and Atoms

Which elements in a direct product $\prod _{\lambda \in \Lambda} R_\lambda$ of rings are atoms? Prove your answer. (Assume this is a commutative ring) My Answer: Atoms of $\prod R_\lambda$ are ...
2
votes
1answer
265 views

Endomorphism Ring Isomorphism

Suppose that $D$ is a division ring and let $M_n(D)$ be the $n\times n$ matrix ring with entries from $D$. $D^n$ is a left module over $M_n(D)$. I want to show that $D$ and End$_{M_n(D)}(D)$ are ...
4
votes
1answer
275 views

Exercise 2.11 Atiyah-Macdonald [duplicate]

This is part of the exercise, I'm stuck with it. $A$ is a commutative ring with unit. 1) Suppose we have an homomorphism $\phi : A^{m} \to A^{n}$ surjective. Is true that $m \geq n $ ? 2) Suppose ...
0
votes
1answer
76 views

Equality with powers of an ideal

Let $A$ be an arbitrary (commutative with an identity) ring. Suppose $\alpha$ is an ideal. Is it true that $$\alpha(\alpha\cap\alpha^2\cap\alpha^3\cap…)=\alpha\cap\alpha^2\cap\alpha^3\cap…?$$ ...
2
votes
1answer
338 views

Endomorphisms of direct sum and division ring

How to prove that $$\operatorname{End}_R(V^{ \oplus n }) \cong M_n(D),$$ where $V$ is a simple left $R$-module and $D=\operatorname{End}_R(V)$. This is part of the proof finally leads to prove ...
1
vote
1answer
70 views

Exercise from Atiyah about flatness

This is an exercise from Atiyah. Let $N$ be a flat $B$-module, and $B$ a flat $A$-algebra where $A$ is a commutative ring with unit. Then $N$ is flat as $A$-module Any hint ?
0
votes
1answer
228 views

Maximal ideal generated by irreducible element

Let $R$ be an integral domain and let $(c)$ be a non-zero maximal ideal in $R$. Prove that $c$ is an irreducible element.
0
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1answer
92 views

Is the homomorphic image of a G-domain is G-domain?

I have no idea how to prove this if it is true or to give a counter example if it is not true. Is the homomorphic image of a G-domain is a G-domain? A G-domain is an integral domain $R$ with ...
3
votes
1answer
194 views

In a Morita context ($A, B, P, Q, f, g$), why are $P$ and $Q$ projective if $f$ is surjective?

The title probably says it all :). This is probably a very very simple question, please bear with me. Let $(A, B, P, Q, f, g)$ be a Morita context (or pre-equivalence data) as defined by Hyman Bass in ...
2
votes
1answer
93 views

In a commutative ring without identity, is $(a)(b)\subset (ab)$ or $(ab)\subset (a)(b)$?

Let $R$ be a commutative ring without unity. Consider an ideal $(a)$ generated by $a\in R$. Note that $(a)=\{ra+na : r\in R, n\in \textbf Z\}$ since $R$ has no identity. I wonder if $(a)(b)\subset ...
0
votes
1answer
61 views

Principal Ideal Domains

I'm trying to teach myself by doing questions. I understand the definition of a ideal is a multiplicatively closed additive subgroup of a ring. And a principal ideal means it has a generator 'g'. So ...
3
votes
2answers
42 views

PID question in Ireland and Rosen

Context: In Ireland and Rosen's 'A classic introduction to number theory' on page 11, the proof that in a PID$=R$, there is an integer $n$ such that, for a prime $p$ and any $b\in R$, $p^n \mid b , ...
1
vote
1answer
32 views

A doubt regarding splitting field.

$\Bbb Q(\omega)= \Bbb Q(\sqrt3,\iota)$ This is written in my text book that i am following. But I think this is a typo. Since $\Bbb Q(\sqrt3,\iota)$ is a larger field. in which $\Bbb Q(\omega)$ is ...
3
votes
1answer
72 views

Classify abelian groups $A$ which are irreducible $End(A)$-modules

Classify abelian groups $A$ which are irreducible $End(A)$-modules. I think i did it for finite abelian group $A$ . A finite abelian group $A$ is irreducible iff order of $A$ a is power of prime. ...
1
vote
2answers
93 views

Example of Localization and Prime Ideals

For each $n\in \mathbb Z^+$, give an example of a localization of $\mathbb Z$ with exactly $n$ prime ideals. Justify your answer. Could an example have something to do with a UFD or Noetherian ...
0
votes
2answers
411 views

Spectrum of a product of rings isomorphic to the product of the spectra

I've found in an exercise this statement: If $A$ is a commutative ring with unit and $A = A_{1} \times \dots \times A_{n}$ then $$\def\Spec{\operatorname{Spec}} \Spec(A) \cong \Spec(A_{1})\times ...
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vote
2answers
31 views

If $R$ is $\text{PID}$ and $x \in R$ is irreducible, then $R/(x^k)$ is a local ring.

Suppose $R$ is $\text{PID}$ and let $x \in R$ be irreducible. Let $k \in \mathbb{Z}_{>0}. $Could anyone advise me on how to prove $R/(x^k)$ has a unique maximal ideal? Hints will suffice, thank ...
1
vote
1answer
101 views

Fraction field of $R/P$

It may be a simple question seeming too easy, but I seek a help: If $P$ is a prime ideal of a commutative ring $R$, could one say that $R_P/PR_P$ is the field of fractions of $R/P$? Thanks a ...
1
vote
2answers
35 views

Is $2\mathbb{Z}_{12}$ maximal ideal of $\mathbb{Z}_{12} \ ?$

I came across this solution to the a/m problem here: https://sg.answers.yahoo.com/question/index?qid=20110515182700AAsSjeA. The author of the solution claims that $\mathbb{Z}_{12}/2\mathbb{Z}_{12} ...
7
votes
2answers
253 views

Why study integrality?

Here are a few of the basic definitions related to integrality. (1) A polynomial in $R[x]$ is monic if its leading coefficient is $1$. (2) An element is integral over a ring $R$ if it ...
0
votes
1answer
23 views

An attempt to verify if $\mathbb{Z}_{7^5}$ a local ring with unique maximal ideal $(7) \ .$

A commutative ring $R$ with identity is called a local ring if there exists unique maximal ideal in $R.$ Hence, is $\mathbb{Z}_{7^5}$ a local ring with unique maximal ideal $(7) \ ?$ Here is my ...
5
votes
1answer
122 views

Finding the conjugates in $\mathbb{C}$ of a given number over a given field…

I'm having somewhat of a difficult time understand what's being asked—and thus having a hard time answering the question: Find all conjugates in $\mathbb{C}$ of the given number over the given ...
0
votes
1answer
64 views

Definition of modules

The definition of modules confuses me: $R$ is a ring, then a left $R$ module is an abelian group $V$ together with a multiplication map $$R \times V \to V, (r,v) \to rv$$ satisfies some natural ...
1
vote
1answer
49 views

Rings act on modules

About the definition of modules: $R$ is a ring, then a left $R$ module is an abelian group $V$ together with a multiplication map $$R \times V \to V, (r,v) \to rv$$ satisfies some natural axioms. I ...
5
votes
2answers
140 views

Generality of rings' abelian group

Let G be an abelian (finite) group. Is there a ring $R$ with $G$ isomorphic to the group $(R,+)$?
3
votes
2answers
51 views

Suppose $R$ is a ring and $\exists n \in \mathbb{Z}_{> 0}$ such that $(ab)^n=ab, \forall a,b \in R.$ Then $ab = 0 $ iff $ba=0 \ ?$

Suppose $R$ is a ring and there exists $n \in \mathbb{Z}_{> 0}$ such that $(ab)^n=ab, \forall a,b \in R.$ Could anyone advise me on how to prove/disprove that $ab = 0$ iff $ba=0, \forall a,b \in R ...
4
votes
0answers
158 views

Why is the completion of the ring of germs of smooth functions $\cong \mathbb{R}[|T|]$?

Let $C^{\infty}$ be the canonical commutative ring on the set $\{f: \mathbb{R} \rightarrow \mathbb{R} \mid f \text{ smooth}\}$. Let $\mathfrak{m}= \{ f \mid f(0)=0 \}$ a maximal ideal. Consider the ...
1
vote
2answers
49 views

Size of an ideal in a polynomial Ring

Let $F$ be a field and let $I = \{f(x) \in F[x]\mid f(a) = 0 ~~ \forall a \in F\}$. Prove that $I = \{0\}$ when $F$ is infinite. I have already shown that $I$ is an ideal and that $I$ is infinite ...
1
vote
1answer
418 views

Nilpotent Elements and Intersection of Prime Ideals

Prove that the set of nilpotent elements of a ring is the intersection of its prime ideals. I know these two useful facts: {nilpotent elements}$=\sqrt{0}$ $\sqrt{I}= \bigcap$ of prime ideals ...
1
vote
1answer
150 views

Ideals and prime ideals in a commutative ring. [closed]

Let $A_1$ and $A_2$ be two ideals, and $P_1$ and $P_2$ be two prime ideals in a commutative ring $R$. Assume that $A_1 ∩ A_2 ⊆ P_1 ∩ P_2$. Is there at least an $i$ and $j$ such that $A_i ⊆ P_j$ is ...
0
votes
1answer
63 views

Show an Ideal is the principal ideal for some polynomial.

Let $F$ be a field and $R = F[X]$. Suppose $I$ is an ideal of $R$. Show that $I = (p(X))$ for some $p(X)$ in $F[X]$. (Hint: consider a polynomial $p(X)$ of least degree in $I$.) I'm trying to do this ...
1
vote
4answers
104 views

Irreducible in $(\mathbb{Z}[i])[x]$

Prove that $p(x)=x^3-6x^2+4ix+1+3i$ is irreducible in $(\mathbb{Z}[i])[x]$. I'm not so clear on irreducibility in specific instances like this one. I know I need to show that if $p(x)=q(x)r(x)$ then ...
2
votes
1answer
179 views

finite k-vector space noetherian

Let $k$ be a field and $A \supset k$ a ring that is finite dimensional as a $k$-vector space, prove that A is Noetherian and Artinian. For the first part, I've been trying to use a.a.c condition, so ...
0
votes
2answers
47 views

Set or Ring, and group of units?

I have a couple of questions. I understand the axioms needed for a ring. But am confused about a unitary ring? does this just mean its a ring but has to have the unit 1? Also I do not understand ...
3
votes
2answers
59 views

Showing that $\mathbb{Z}/2\mathbb{Z}[\mathbb{Z}/2\mathbb{Z}]$ is semi-simple

Showing that $\mathbb{Z}/2\mathbb{Z}[\mathbb{Z}/2\mathbb{Z}]$ is semi-simple. Then I denote that first $\mathbb{Z}/2\mathbb{Z}=\{0,1\}$ and the second one as $\{e,g\}$. Then I have that ...
4
votes
2answers
108 views

Books on Rings without Identity

I was just wondering if anybody knows of any good books or articles that study rings (and algebras) without (or not necessarily with) identity. I have gone through Thomas Hungerford's Algebra textbook ...
0
votes
0answers
102 views

Krull's theorem and AC

Lately I have been trying to prove axiom of choice based on krull theorem. The theorem states that for every ring with a unit $R$ that is not a field, there is a maximal ideal. I know it is equivalent ...
2
votes
1answer
114 views

Is my proof for this completion of a ring not being flat correct?

I wanted to show that for $A = K[X_i, i \in \mathbb{N}]/(X_iX_j)_{i,j \in \mathbb{N}}$ the completion $A[T] \rightarrow A[|T|]$ is not flat. However, my proof seems a bit simple/ direct to me, so I'm ...
3
votes
1answer
342 views

Prove that R is a field.

I have a question: Let R be an integral domain, if the only ideals are the trivial ideals, prove that R is a field. So my proof for this was; Let $a\in R$ be a non-zero element. Since we have that ...
2
votes
1answer
106 views

Example of a finite ring with identity containing a ring without identity

What is an example of a finite ring $R$ with unity and a subring $S$ of $R$ that is not a ring with unity?
0
votes
1answer
241 views

Let R be a commutative ring with identity. Prove that R is a field if and only if (0R) is a maximal ideal. [duplicate]

I'm confused on how to go in both directions and how to start this proof.
0
votes
1answer
42 views

Polynomial rings not useful when $R$ is a field?

When $R$ is a field then surely we end up for no use for polynomial rings as $R[x]=R$? As you can prove that: Suppose that $R$ is an integral domain. Then $(R[x])^{\times}=R^{\times}$ This ...
0
votes
1answer
75 views

Noetherian ring of Krull dimension $0$

I've found this claim: Let $A$ be a Noetherian ring of Krull dimension $0$ . Then $A$ is a field or it has a finite number of prime ideals. Why is this true ?
-2
votes
1answer
100 views

prime ideals contains comaximal

Let $R$ be a commutative ring with unity 1 and $I$, $J$ and $P$ ideals in $R$ show that if every prime ideal of $R$ contains either $I$ or $J$ ,but not both then $I$ and $J$ are comaximal ...
-1
votes
1answer
49 views

integral over proof

Let $S$ be a sub ring of a commutative ring $R$ let $y$ be in $R$. Prove that if $y$ is integral over $S$, then $S[y]$ is integral over $S$. I'm puzzled as how to start this one.
-1
votes
1answer
98 views

Irreducibility over $ \mathbb{Q} ( \sqrt{2} , \sqrt{3})$ [closed]

Show that $x^5-9 x^3 +15x +6$ is irreducible over $ \mathbb{Q} ( \sqrt{2}, \sqrt{3})$