0
votes
0answers
9 views

Morandi's Rings appendix: about a step of the proof that that $R[x]$ is a UFD if $R$ is. [duplicate]

In the appendix about rings in Patrick Morandi's book Fields and Galois Theory, we find the following exercise (which arises in the proof of the theorem: $R[x]$ is a UFD if $R$ is a UFD). Let B be ...
0
votes
0answers
28 views

Subring of $\mathbb{Z}[i]$ and an infinite set $X$ such that $\exists x \forall y \in X \,\,x^2\mid y^2$ but $\forall x \forall y \,\,x \not\mid y$

This is a question derived from A subring of the ring of Gaussian integers such that $a^2 \mid b^2$ does not lead to $a\mid b$ in infinitely many such cases. Is there a subring $R$ of Gaussian ...
0
votes
1answer
45 views

A subring of the ring of Gaussian integers such that $a^2 \mid b^2$ does not lead to $a\mid b$ in infinitely many such cases

As the ring of Gaussian integers is a UFD, this means that $a^2 \mid b^2$ leads to $a\mid b$. Is there any subring of the ring of Gaussian integers with infinitely many elements such that ...
3
votes
3answers
101 views

Prove that the division ring is commutative if for every $x$, $x^7=x$

I'm trying to solve a problem and I'm stuck. Here is the original problem: Let $A$ be a finite-dimensional algebra over a field $K$, such that for every $a\in A$, $a^7=a$. Show that $A$ is a ...
0
votes
1answer
20 views

Semiring that has unique factorization except zero

In a ring, there is unique factorization domain. Then is there a similar concept in semiring - that is a commutative semiring that has unique factorization for every element except zero? If so, what ...
1
vote
3answers
57 views

Prove that $I$ is a maximal ideal of $\mathcal A$. [duplicate]

Please, give-me a hint to prove this proposition: Let $\mathcal A$ be the ring of all continuous real functions (with the usual operations of sum and multiplication) defined on the interval ...
0
votes
0answers
27 views

An integral domain that has square of prime elements share same greatest common factor, whil [on hold]

Is there any numerical integral domain, not involving monomials or polynomials that has square of prime elements share same greatest common factor $g$, while product $P$ of two different prime ...
-1
votes
3answers
62 views

Annihilator of annihilator of annihilator of a submodule

Let $R$ be a commutative ring and M be an $R$-module. The question is whether for every submodule $N$ of $M$ the equality ...
2
votes
1answer
58 views

How to split up $X^{20}-1$ in $\mathbb{F}_3[X]$

I try to split $X^{20}-1$ into irreducible polynomials in $\mathbb{F}_3[X]$. The first thing I saw is that $1$ is a root. Second, $-1$ must be one too. I have taken the derivative $20X^{19}$ to ...
1
vote
0answers
26 views

Soluble group algebras and centralizers

Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and let $H$ be ...
2
votes
1answer
49 views

Valuation rings of $k(X)$

My question is how to determine all valuation rings of the field $k(X)$ containg the field $k$. I want to show that if $V$ is a valuation ring of the field $k(X)$ and $\neq k(X)$ then ...
6
votes
4answers
143 views

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$?

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$? I wrote it down in an imprecise way on purpose. The notation above is the linear algebra one: ...
0
votes
1answer
31 views

Maximal ideal in the ring of continuous functions [duplicate]

Let $R$ be the ring of continuous functions $\mathbb{R}\rightarrow\mathbb{R}$ with the usual operations and $I$ the subset of functions $f$ with $f(x_0)=0$ for some $x_0\in\mathbb{R}$. It's easy to ...
0
votes
0answers
19 views

There exists only a finite number of ideal classes in a number ring

Let $K$ be a number field (i.e. $\mathbb Q\le K\le\mathbb C$ s.t. $[K:\mathbb Q]=n$) and $R=\mathbb A\cap K$ the relative number ring. Calling $\Phi(R)$ the set of ideals of $R$, we define on it the ...
-5
votes
0answers
29 views

How I can find the Jacobson radical? [on hold]

Need answer for this question in rings theory; find J(M2(R)) If R=\begin{pmatrix} z/12z& 2z/12z\\ 0& 3z/12z\\ \end{pmatrix} Where J denoted the Jacobson radical. Thanks.
0
votes
2answers
43 views

Nilpotent element in commutative ring [duplicate]

Let $A$ be a commutative ring, prove that if $x \in A$ is nilpotent then $1-x$ is an invertible element in $A$. I need help with this one.
0
votes
0answers
31 views

Kernel of a homomorphism: why $g_i(\alpha)\in Q_i$?

Let $K\le L$ be two number fields, $[L:K]=n$. Let $R=\mathbb A\cap K$ and $S=\mathbb A\cap L$ be the relative number rings. Take $\alpha\in S$ an element of degree $n$, i.e. such that $L=K[\alpha]$. ...
0
votes
0answers
38 views

If field has a prime field isomorphic to $\mathbb{Q}$, sufficient condition for every subring being integrally closed domain

Suppose that a field $k$ has the prime field isomorphic to the field of rational numbers $\mathbb{Q}$. Then what would be sufficient condition in order for every subring of $k$ be integrally closed ...
1
vote
0answers
32 views

When is a subring of a field an integrally closed domain? [on hold]

What criteria would be necessary/sufficient for a subring of a field to be an integrally closed domain?
4
votes
0answers
57 views

Condition on a field that makes every subring an integrally closed domain

I want to know what condition would need to be additionally imposed on a field to make every subring of the field an integrally closed domain.
0
votes
2answers
48 views

What would be a prime element in the field of rational numbers?

It is clear to understand what prime elements will be in case of ring of integers and many other rings. However, I find it confusing in case of fields, more specifically field of rational numbers. So ...
0
votes
0answers
20 views

A non-UFD where there exist infinitely many elements such that $a^2 \mid b^2$ does not lead to $a\mid b$ [duplicate]

Is there a commutative non-$\text{UFD}$ ring such that there exists a set $X$ of infinite cardinality of elements that for $\forall x \in X$, $x^2$ is a multiple of $a^2$ for some particular $a$, but ...
0
votes
1answer
31 views

Example of commutative algebra over integers where there exists $x$ such that $x = y^2$ for several $y$'s

Is there a commutative algebra over integers such that there exists $x$ with $x = y^2$ for several $y$'s? Also, is there a commutative algebra over integers such that for every $k \in \mathbb{N}$, ...
7
votes
1answer
61 views

A central division algebra is not its commutator

In looking at old qualifying exam questions, I've come upon a question that has me stumped. Let $A$ be a central division algebra (of finite dimension) over a field $k$. Let $[A,A]$ be the ...
0
votes
1answer
34 views

Relating the characteristic of the ring R to the characteristic of R[x]

Suppose $R$ is a ring and $R[x]$ is the ring of polynomials in the indeterminate $x$ with coefficients from $R$. The characteristic of a ring is the smallest positive integer $n$ such that $n \cdot r ...
1
vote
0answers
50 views

Rapid and easy question on ideals and ring

Let $R$ be the number ring related to a field $K$ of finite degree over $\mathbb Q$, i.e. $\mathbb Q\le K\le\mathbb C$ and $[K:\mathbb Q]=n$. Hence $R=\mathbb A\cap K$, where $\mathbb A$ is the ring ...
3
votes
0answers
35 views

Non-UFD that there exists set $X$ of any cardinality and $a$ that $xy$ is not divisible by $a^2$ for any $x,y \in X$ and $x^2$ is divisible by $a^2$

Let's say we want to construct a non-UFD that is a commutative ring that satisfies: There exists $a$ such that there exists a set $X$ of elements of finite cardinality $k$ such that for any $x,y \in ...
0
votes
3answers
41 views

Converting a polynomial ring to a numerical ring

One may be curious why one wishes to convert a polynomial ring to a numerical ring. But as one of the most natural number system is integers, and many properties of rings can be easily understood in ...
4
votes
2answers
80 views

A non-UFD such that $a^2 \mid b^2$ does not lead to $a\mid b$

Is there any non-UFD that is a commutative ring such that $a^2 \mid b^2$ does not always lead to $a\mid b$? It would be preferable if examples are something that does not involve ...
1
vote
0answers
35 views

Showing the existence of sub fields of a finite field

For each divisor $m$ of $n$, $GF(p^n)$ has a unique sub field of order $p^m$ . The proof of this theorem in Gallian goes like this : Suppose that $m$ divides $n$. Then, since : $(p^n-1) = ...
3
votes
3answers
98 views

Is division allowed in rings and fields?

Is division allowed in ring and field? The definition of ring I am using here does not require the presence of multiplicative inverse. I think in general, division is not a well-defined ...
1
vote
3answers
58 views

A number system that is not unique factorization domain

Can anyone present a number system that is not unique factorization domain and is a commutative ring? So I want the case that does not involve polynomials/monomials or some trivial cases.
0
votes
1answer
53 views

Proof that presheaf is a sheaf for Spec

Atiyah Macdonald define presheaf (chapter 3, exercise 23) on the base of $Spec(A)$, where $A$ is commutative ring with $1$, as follows $$ \mathfrak{F}(X_f) = A_f, $$ where $X_f$ is a basic open set ...
-3
votes
2answers
71 views

Idempotent elements of a ring.

I need the idempotent elements of $Z_{900}$ $2^2\cdot 3^2\cdot 5^2=900$ Of course there's $$0 \pmod 4 \\ 0 \pmod 9 \\ 0 \pmod {25} \\ $$ and $$ 1 \pmod 4 \\ 1 \pmod 9 \\ 1 \pmod {25} \\ $$ I found ...
7
votes
3answers
185 views

In $\mathbb{Z}/(n)$, does $(a) = (b)$ imply that $a$ and $b$ are associates?

[Update: Based on the hints provided by @zcn and @whacka, I believe I have found a solution. See my answer below.] Below, $R$ is a commutative ring with $1$. In John J. Watkins' Topics in ...
2
votes
1answer
48 views

Von Neumann regular but not self-injective ring

I want an example of a von Neumann regular ring which is not self-injective. My thanks go to anybody answering.
3
votes
2answers
142 views

Sum and Product of two transcendental numbers cannot be simultaneously algebraic

If $\alpha$ and $\beta$ are real number and $\alpha$ and $\beta$ are transcendental over $\mathbb Q$, show that $\alpha \beta$ or $\alpha +\beta$ is also transcendental over $\mathbb Q$ Attempt: ...
1
vote
1answer
40 views

Characterizing Prime and Maximal Ideals in a nice Ring

Consider the "nice" ring $(\mathbb{Z}/20\mathbb{Z})[x]$ and I am trying to list all the prime and maximal ideals of this. The reason I call this a nice (or manageable) ring is because we ...
2
votes
1answer
33 views

Example of a ring which is not CM at all its prime ideals

A commutative ring $A$ is said to be CM at a maximal ideal $\mathfrak{m}$ if and only if $Depth(A_{\mathfrak{m}})=Krull(A_{\mathfrak{m}})$. What is an example of a connected commutative ring $A$ which ...
0
votes
1answer
48 views

A nonregular local ring [duplicate]

Consider the ring of the formal power series $k[[T_1,\ldots,T_n]]$ ($k$ algebraically closed) where $\mathfrak m$ is the maximal ideal. If $f\in\mathfrak m^2$, why $$\frac{k[[T_1,\ldots,T_n]]}{(f)}$$ ...
1
vote
2answers
118 views

Is there a more direct way of proving that this ring is an integral domain?

In self studying abstract algebra and I've come upon the following problem which I could not solve directly. For any $d\in \mathbb{Z}$ we are asked to show that $\mathbb{Z}[\sqrt d]=\{a+b\sqrt{d} ...
0
votes
1answer
73 views

Some residue field

Consider a prime ideal $\mathfrak{p}\in\mathrm{Spec} \ \mathbf{Z}[x]$; the residue field at $\mathfrak{p}$ is the fraction field of $\mathbf{Z}[x]/\mathfrak{p}$. Can we classify the residue fields? I ...
4
votes
1answer
35 views

Let $K$ be a field extension of $F$ and let $a \in K$. Show that $[F(a):F(a^3)] \leq 3$

Let $K$ be a field extension of $F$ and let $a \in K$. Show that $[F(a):F(a^3)] \leq 3$. Find examples to illustrate that $[F(a):F(a^3)]$ can be $1,2$ or $3$. Attempt: $F \subset F(a^3) \subseteq ...
0
votes
0answers
45 views

A non-unital commutative ring with infinite elements such that each element $a$ satisfies $ab =0$ for infinitely many $b$'s

Beside the usual rules for non-unital commutative ring (that is, a ring without multiplicative identity) $R$, I want $R$ to satisfy the following: $ab = 0$ for each element $a$ and there are ...
0
votes
0answers
108 views

Multiplicity of a module in a particular case

We define multiplicity of a module M of dimension $d>0$ as $$mult(M) := lc (P_M) (d-1)!$$ where $P_M$ denotes the Hilbert polynomial of M. Equivalently, we have $mult(M) = Q_M(1)$, where $HP_M (z) ...
2
votes
1answer
23 views

A field extension of prime degree

Suppose that $E$ is an extension of $F$ of prime degree. Show that $~~\forall~ a \in E : ~ F(a)=F$ or $F(a)=E$ Attempt: Suppose that $E$ is an extension of a field $F$ of prime degree, $p$. ...
2
votes
1answer
30 views

Suppose that $f(x)$ and $g(x)$ are irreducible over $F$ and that $\deg f(x)$ and $\deg g(x)$ are relatively prime.

Suppose that $f(x)$ and $g(x)$ are irreducible over $F$ and that $\gcd(~\deg g(x),\deg f(x)~)=1$. If $a$ is a zero of $f(x)$ in some extension of $F$, show that $g(x)$ is irreducible over $F(a)$ ...
1
vote
1answer
37 views

Krull's theorem for a ring that does not have unit (multiplicative identitiy)

Is there some sorts of Krull's theorem (that every ring has maximal ideal) for rings that do not have multiplicative identity (unit)? So I know that non-unital rings do not satisfy Krull's theorem, ...
2
votes
2answers
29 views

Nil radical of an ideal on a commutative ring

This is a problem of an exercise list: Let $J$ be an ideal of a commutative ring A. Show that $N(N(J))=N(J)$, where $N(J)=\{a \in A; a^n \in J$ for some $n \in \mathbb{N}\}$. What I did: ...
0
votes
2answers
60 views

Cohen-Macaulay and regularity

I know this is a simple question but to make sure....: $A$ is a commutative ring which is Cohen-Macaulay and for every maximal ideal $\mathfrak{m}$ in $A$ if $\dim A_{\mathfrak{m}}=\dim A$ then ...