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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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
97 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
vote
1answer
48 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
1k 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
77 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 ...
4
votes
2answers
253 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 ...
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0answers
212 views

Example of integral domain with infinitely ascending chain of ideals. [duplicate]

I am looking for an integral domain in which we have an infinitely ascending chain of ideals. Clearly, this can't be a PID. Also, I am looking for examples other than infinite dimensional fields, ...
2
votes
1answer
314 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
91 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
55 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
104 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
98 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( ...
-1
votes
1answer
72 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
32 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
157 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
363 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
68 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!!
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votes
7answers
506 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
54 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
63 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 ...
4
votes
2answers
162 views

Localisation and prime ideals

If $A$ is a ring and $S=\{1,f,f^2,f^3,...\}$ a multiplicative set of $A$. Prove that $Spec(A_f)=(\mathfrak{V}((f)))^c$. Notation: $A_f=S^{-1}A$ and $\mathfrak{V}((f))=\{P \in Spec(A): P \supset (f)\}$ ...
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vote
0answers
35 views

Functional equation on a ring

Let $R$ be a ring with identity. Find all functions $f: R \to \Bbb R$ satisfying the functional equation \begin{equation} f(xp+yq, yp+xq)=f(x, y)f(p, q) \end{equation} for all $x, y, p, q\in R$, and ...
0
votes
1answer
119 views

$(p)$ is a prime ideal in $\mathbb{Z}[\sqrt{n}]$ if and only if $n$ is not a square mod $p$

Let $n$ be a square-free integer such that $n\equiv 2,$ or $3\bmod 4$. If $p\in\mathbb{Z}$ be a prime such that $n$ is not a square modulo $p$, then $(p)$ is a prime ideal in ...
1
vote
2answers
299 views

How to show $\mathbb{Z}[x]/<2,x>$ is isomorphic to $Z_2$

I'm having quite a bit of trouble figuring out why $\mathbb{Z}[x]/<2,x>$ is isomorphic to $\mathbb{Z}_2$. So far I have figured out there is an onto map $\zeta: ...
0
votes
3answers
179 views

Showing quotient rings are isomorphic

Can anyone explain to me how to show two quotient rings are isomorphic? For my particular case. Both quotient rings are based off ideals in the ring $\mathbb Z_3[X]$: $$ \mathbb ...
0
votes
0answers
39 views

Why an element of an order of a number field K is always an algebraic integer of K?

Let $K=\mathbb Q(\sqrt{N})$ be a number field, $\mathcal O$ be an order of $K$ (i.e. $\mathcal O$ is a subring of $K$ and $\mathcal O$ is a free $\mathbb Z$-module of rank 2). In the begining of ...
1
vote
1answer
458 views

Unique prime ideal implies every element is nilpotent or a unit.

Let $R$ be a commutative ring with only one prime ideal. I want to show that every element of $R$ is either a unit or nilpotent, or equivalently, that the nilradical is the unique maximal/prime ideal. ...
3
votes
1answer
80 views

Kernel of a morphism of regular rings.

Let $k$ be a field and $f: A \rightarrow B$ be a surjective ring morphism between smooth Noetherian $k$-algebras. By smooth I mean that the module of Kahler Differentials $\Omega_{A|k}$ is a ...
2
votes
3answers
137 views

A homomorphism induces a continuous map from ${\rm Spec}(A') \to {\rm Spec}(A)$.

Let $A, A'$ be commutative rings with $1 \neq 0$. Let $h : A \to A'$ be such that $h(1) = 1$. Then $f: {\rm Spec}(A') \to {\rm Spec}(A)$ defined by $f(\mathfrak{p}') = h^{-1}(\mathfrak{p}')$ is ...
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vote
0answers
58 views

Module is free of finite rank $\implies$ submodule is free of finite rank?

Let $M$ be $R$-module, where $R$ is commutative ring with $1,$ and $N$ be submodule of $M.$ If $M$ is free of finite rank, so is $N \ ?$ Answer: False. Let $R=M=\mathbb{Z}_{6}$ and ...
1
vote
1answer
27 views

Polynomial ring problem

May I verify if my proof to this problem is correct? Let $p \in \mathbb{P}.$ For $x \in \mathbb{Z},$ let $\overline{x}$ be remainder of $x$ when divided by $p.$ Let $f(X)= \sum^{n}_{i=0}a_iX^i ...
2
votes
2answers
200 views

Irreducible components of $Spec(A) $

A topological space $X$ is called irreducible if given $A_{1}, A_{2} $ open sets $ \neq \emptyset $ then $A_{1} \cap A_{2} \neq \emptyset$. The maximal irreducible topological subspaces of $X$ are ...
0
votes
1answer
32 views

Annihilaor of a prime is non-zero

If we have a commutative noetherian ring how do we know that the annihilator of a prime ideal is always non-zero?
1
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1answer
41 views

A problem on non-commutative ring

Let $R$ be a non-commutative ring with $1$ and $a,b\in R$ such that $ab=1 \neq ba.$ Could anyone advise me on how to show there exists $c\in R-\{b\}$ such that $ac=1 \ ?$ Hints will suffice. Thank ...
0
votes
1answer
49 views

Eisenstein's criterion pf

I know that 'Eisenstein's criterion'. I know that pf of state "(NOT $p$|$a_{n}$), [$P|a_i$ for ($0\le i\le n-1$)], (NOT $p^2$|$a_0$)". I know regular way. but I hope to Second pf way. $\;$ $\;$ ...
0
votes
1answer
393 views

Formal Power Series Ring, Maximal Ideal

Let K be a field. Show that K[[x]] (the formal power series ring with coefficients in K) has a unique maximal ideal. Attempt at a solution: Let I $\subset$ K[[x]] with I $\neq$ K[[x]] and I= ($x$). ...
3
votes
1answer
188 views

If $M$ is Noetherian, then $R/\text{Ann}(M)$ is Noetherian, where $M$ is an $R$-module

Let $M$ be an $R$-module and $\text{Ann}(M)=\{r \in R: rm =0 , \forall m \in M\}.$ Suppose $M$ is Noetherian. Could anyone advise me on how to prove $R/\text{Ann}(M)$ is also Noetherian? Hints will ...
1
vote
1answer
96 views

Noncommutative finitely generated algebras need not be noetherian

I would like to understand an example (of the title) given in the book "An Introduction to Noncommutative Noetherian Rings" by K. R. Goodearl, R. B. Warfield... On page 8, Exercise 1E, an example of ...
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votes
1answer
49 views

$f(x)$ is irreducible and primitive over $\Bbb Q$, then $f(x)$ is irreducible over $\Bbb Z$

show that $f(x) (\in \Bbb Z[x])$ is irreducible and primitive over $\Bbb Q$, then $f(x)$ is irreducible over $\Bbb Z$. How pf it? I tried it. MY pf) Suppose that $f(x)$ is reducible over $\Bbb Z$. ...
1
vote
1answer
63 views

An induction proof on a version of “prime avoidance” from Atiyah-McDonald.

The proposition is 1.11 from the commutative algebra book. Let $\mathfrak{p}_1, \dots, \mathfrak{p}_n$ be prime ideals and let $\mathfrak{a}$ be an ideal contained in the union of those prime ideals. ...
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votes
1answer
2k views

Online Finite Field Calculator

I need to find an online Finite Field calculator with the following operations: Inverse SqrRoot Mult Div I have found one a couple of days ago but lost the url, and cannot find it now. Any ...
0
votes
1answer
38 views

$V(\mathfrak{a} \cap \mathfrak{b}) = V(\mathfrak{a}) \cup V(\mathfrak{b})$ (Spectrum of a commutative ring)

Let $V(\mathfrak{a})$ be all ideals in ${\rm Spec}(A)$ that contain ideal $\mathfrak{a}$. Then $V(\mathfrak{a} \cap \mathfrak{b}) = V(\mathfrak{a}) \cup V(\mathfrak{b})$. $\mathfrak{p} \in$ RHS ...
1
vote
1answer
49 views

How to show $c-b\lt b-a$

The question: Let $G$ be an Arf semigroup and $a\lt b\lt c$ be three consecutive elements in $G$. How to show that $c-b\lt b-a$ and how to show that this is not necessarily the case for every ...
0
votes
1answer
413 views

Sylvester domains

I'm an undergrad mathematics student and I'd like to request some books about Sylvester domains. Specifically I'd like to understand the fact that not all modules are Sylvester domains. I just proved ...
3
votes
1answer
134 views

Global dimension of $\mathbb Q [x]$

I'm trying to show that the global dimension of $\mathbb Q [x]$ is $1$. I have shown that $D(\mathbb Q [x]) \leq 1$ as follows. One can reduce to the case of showing that ...
0
votes
1answer
56 views

Questions on a proof of “All prime ideals of a Dedekind domain are invertible”

I tried to prove this theorem : All prime ideals of a Dedekind domain is invertible. i.e, For every prime ideal $\mathfrak{p}$ of Dedekind domain $R$, there exists $\mathfrak{p}^{-1} \subseteq ...
0
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1answer
76 views

$\Bbb Z[i\alpha]$ UFD's

I know that $\Bbb Z[i]$ and $\Bbb Z[\sqrt{-2}]$ are Unique Factorization Domains, and that $\Bbb Z[\sqrt{-6}]$ is not. I have two questions. I know that they may be difficult questions, so I only ask ...
1
vote
1answer
58 views

Proving that $V(R^*)=V(R)-1$

Let $R$ be a noetherian local ring with Jacobson radical $J$. Define $V(R)=\dim J/J^2$ where $J/J^2$ is considered as a vector space over $R/J$. Now fix $x\in J-J^2$. If we then let $R^*=R/xR$ then ...