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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0
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2answers
159 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 ...
1
vote
2answers
29 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
89 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
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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
217 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
21 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 ...
4
votes
1answer
78 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
47 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
44 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
136 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
50 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 ...
3
votes
0answers
107 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
38 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
199 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
132 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
55 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
100 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
101 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
44 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 ...
3
votes
1answer
73 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 $Algebra$ ...
0
votes
0answers
83 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
99 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
116 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
80 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
126 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
41 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
59 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 ?
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1answer
76 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.
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votes
1answer
85 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})$
1
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1answer
44 views

Let $\alpha \in \mathbb{C}$. Show that $\mathbb{Z}[\alpha] = \{f(\alpha):f \in \mathbb{Z}[x]\}$

I have been given this problem to prove: Let $\alpha \in \mathbb{C}$ and $\mathbb{Z}[\alpha]$ be the intersection of all unital subrings of $\mathbb{C}$ containing $\alpha$. Show that ...
2
votes
1answer
470 views

Irreducible elements in a PID are prime

How can I see that all irreducible elements in a principal ideal domain are prime? $u$ is irreducible when $u_1 u_2 = u \implies u_1 $ or $u_2$ is a unit. $u$ is prime when $u | ab \implies u|a$ or ...
2
votes
1answer
70 views

Determine up to isomorphism all semisimple noncommutative rings with order 512

Problem: Determine up to isomorphism all semisimple noncommutative rings of order 512 = $2^9$. (This is problem from an old qualifying exam I am studying from) So far I have: Let A be a semisimple ...
3
votes
2answers
149 views

Relationship between the characteristic of a ring and a quotient ring

Let $A$ be a ring with unity such that $\operatorname{char} A=8$. Let $I$ be an ideal of $A$. Show that $\operatorname{char}(A/I) \neq 0$ and $\operatorname{char}(A/I)\leq 8$. What I think I know: ...
4
votes
1answer
67 views

Can something divide one of its divisors?

Let $x$ be an element of a ring and $d$ a divisor of $x$. Can we have $x \mid d$? There's the trivial case where both $x$ and $d$ are units. Otherwise, we have $x=da$ and $d=xb$, thus $x=xba$, so the ...
2
votes
1answer
198 views

Direct product of Noetherian rings is noetherian

Let $R_{1}\times R_{2}$ be a direct product of Noetherian rings. Prove the product is Noetherian. An ideal of $R_{1}\times R_{2}$ is of the form $I_{1}\times I_{2}$ where $I_{1}$, $I_{2}$ are ...
0
votes
1answer
82 views

Prove a module is Noetherian

Let $R$ be a commutative Noetherian ring. Prove that an $R$ module $M$ is noetherian iff $M$ is finitely generated. One way is obvious. The other I have to prove every submodule of $M$ is finitely ...
0
votes
1answer
36 views

To show a certain endomorphism ring is an algebra

Let $A$ be an algebra over a field $K$ of finite dimension. Let $M$ be a finitely generated $A$-module. My question is about the ring ${\rm End}_A(M)$ of $A$-endomorphisms of $M$. I want to show that ...
2
votes
1answer
56 views

Projectiveness and Dedekind domains

Let $A$ be a commutative ring with unity and $M$ an $A$-module. Show that $M$ is flat if and only if $M_\mathfrak{m}$ is a flat $A_\mathfrak{m}$-module for all maximal $\mathfrak{m}\subseteq A$. ...
0
votes
1answer
87 views

$\biggl( \prod_{i \in \mathbb{N}}\mathbb{Z}/p^i\mathbb{Z}\biggr)\otimes_{\mathbb{Z}} \mathbb{Q}\ne 0$

I've found this claim $$\biggl( \prod_{i \in \mathbb{N}}\mathbb{Z}/p^i\mathbb{Z}\biggr)\otimes_{\mathbb{Z}} \mathbb{Q} \not\cong \prod_{i \in \mathbb{N}}\biggl( ...
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votes
1answer
40 views

Is a direct sum of two commutative rings still commutative?

Is a direct sum of two commutative rings still commutative?
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votes
1answer
31 views

Is $F$ a homomorphism?

Let $F$ be the function $\mathbb R[x]$ to $\mathbb R$ by the rule $F(p(x)) = p(0)$. Is $F$ a ring homomorphism from $\mathbb R[x]$ to $\mathbb R$?
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votes
2answers
43 views

Is this structure an ideal?

If we view $\mathbb R$ as a subset of $\mathbb R[x]$ by just identifying $r$ with $r + 0x + 0x^2$, etc. I know then that $\mathbb R$ is a subring of $\mathbb R[x]$. But is it an ideal?
3
votes
2answers
126 views

$\text{V}$ is semisimple as a $k[X]$-module if and only if $A\in\operatorname{End}_k(\text{V})$ is diagonalizable over the algebraic closure of $k$

Suppose $\text{V}$ is a finite-dimensional $k$-vector space and let $A\in \operatorname{End}_k\left(\text{V}\right)$. Regard $\text{V}$ as a $k[X]$-module via $f(X)v=f(A)v$ for $f(X)\in k[X]$, ...
4
votes
4answers
237 views

Localisation isomorphic to a quotient of polynomial ring [duplicate]

Let $R$ be a commutative ring and $A=\{1,a,a^2,\dots\}$ for some $a\in R$. Prove that $A^{-1}R$ is isomorphic to $R[T]/(aT-1)$. I guess I'm meant to find a surjective homomorphism between ...
2
votes
3answers
55 views

prove that if $a$ is not a unit in $\mathbb Z$/m$\mathbb Z$ then $a$ is a zero divisor in $\mathbb Z$/m$\mathbb Z$?

Ho can I prove that if $a$ is not a unit in $\mathbb Z$/m$\mathbb Z$ then $a$ is a zero divisor in $\mathbb Z$/m$\mathbb Z$ ? I am stuck on this problem I would appreciate a lot your help thanks!!
8
votes
7answers
358 views

Examples of properties not preserved under homomorphism

An isomorphism indicates that two structures are the same, using different names for the elements. Therefore it's obvious that every (algebraic) property of the first structure must be present in the ...
1
vote
2answers
47 views

Ring Properties [duplicate]

If $(S,.,+)$ is a ring with the property that $a^2 = a$ for all a an element of $S$, which of the following must be true, given: I $a + a = 0$ for all $a\in{ S}$. II $(a + b)^2 = a^2 + b^2 $ for ...
1
vote
1answer
45 views

transitivity of integral extensions

Let $T{\geq}S{\geq}R$ be commutative rings. I'm trying to prove that if $T$ is integral over $S$ and $S$ is integral over $R$ then $T$ is integral over $R$. Let $t$ be in $T$ so there exist ...