This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

learn more… | top users | synonyms (1)

2
votes
2answers
38 views

Example of ring which is neither commutative nor unital

Give an example of ring which is neither commutative nor unital. I think, subring of matrix ring is neither commutative nor has a unit element.
1
vote
1answer
29 views

Explain rings and is [S, /, -] a ring?

Okay, so we are going to use the base set of numbers [i], which contains all possible cases of ai, where a is any real number. Here are 4 possible groups on this set --> [i,*]... [i,+]... [i,/] ...
0
votes
1answer
22 views

Finite rings without unity that are subrings of finite rings with unity

I know that a ring $R$ without unity can be embedded as a subrng of a ring with underlying additive structure $R \oplus \mathbb{Z}$, a ring with unity. But this does not yield a finite field. But I ...
0
votes
0answers
24 views

An explict description of the integral closure of $A=k[x,y]/\langle x^3-y^2\rangle$. [duplicate]

Let $k=\mathbb C$ and $A=k[x,y]/\langle x^3-y^2\rangle$. Denote by $X$ and $Y$ the cosets of $x$ and $y$ in $A$. Question: How do we see that the integral closure $A'$ of $A$ is $k[Y/X]$? Since ...
0
votes
0answers
11 views

Polynomial modulus in Quotient Ring

I have a ring $R=\Bbb Z[x]/(x^m+1)$ with $m$ some power of two and a polynomial $g \in R$, which has relatively small coefficients and some other properties that I believe to be irrelevant for this ...
0
votes
0answers
10 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
31 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
46 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 ...
1
vote
1answer
62 views

Ideal generated by a regular sequence

I need to prove that the ideal $$ I = (xz -y^2, x^2t^2 -yz^3, x^2yt^2 -xz^4) \subset R = \mathbb{K}[x,y,z,t]$$ is generated by a $R$-regular sequence. How can I do it? I don't know if this can ...
0
votes
2answers
49 views

Calculating the kernel of a homomorphism

Let $R := k[x, y]$ be a polynomial ring over field $k$. Consider the homomorphism $\lambda : k[x, y, z] \to R \times R$, defined by $\lambda(x) := (x, x)$, $\lambda(y) := (y, y)$ and $\lambda(z) := ...
3
votes
3answers
103 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
58 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
22 views

Does there exist a finite axiomatization of the quasi-algebraic theory of real matrix rings?

Some definitions. Let us take the signature of ring theory to consist of the function symbols $\{+,-,0,\cdot,1\}$ equipped with their usual airities, where the minus symbol represents a unary ...
2
votes
0answers
36 views

the numer of monic irreducible polynomials of degree $3$ in $\mathbb{F}_q$

I want to know how hany monic irreducible polynomials of degree $3$ there are in a field $\mathbb{F}_q$. The whole number of monic polynomials of degree three is $q^3$. Now I want to find out how ...
1
vote
0answers
27 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
50 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
146 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.
3
votes
2answers
50 views

Show that if $\mathrm{Tr}(y)=0$ then there exists a $x$ such that $x^p-x=y$.

We have the Trace map defined by: $$ \mathrm{Tr}\colon \mathbb{F}_q\rightarrow\mathbb{F}_q\colon x\mapsto x+x^p+x^{p^2}+\cdots+x^{p^{n-1}}, $$ where $q=p^n$. Now I have to prove that if ...
8
votes
2answers
61 views

If $\forall x \in R, x^2-x \in Z(G)$, than $R$ is commutative [duplicate]

Let $R$ be a ring such that for every $x\in R$ we have $x^2-x \in Z(G)$. Show that $R$ is a commutative ring. My thoughts What should I do? I could show that every $y \in R$ could be written in ...
0
votes
0answers
37 views

$I_M=J_M$ for all $M \in \operatorname{Max}(R)$ implies $I=J$ [on hold]

Let $R$ be a commutative ring with unity. Show that if $I_M=J_M$ for all $M \in\operatorname{Max}(R)$ then $I=J$.
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 $L=K[\alpha]$. ...
4
votes
0answers
60 views

Functorial approach to Ideals and Quotients, Multiplicative Sets and Localizations

I have been playing with substructures of commutative rings today and noticed that there is a strong analogy between the formation of quotients and kernels with the formation of localizations with ...
0
votes
0answers
39 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? [closed]

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.
1
vote
0answers
19 views

Show that $31 | ord(\alpha)$ for a root of $f \in \mathbb{F}_{5}$

Let $f$ be an irreducible monical polynomial of in $\mathbb{F}_5[X]$ such that $\deg(f)=3$, and let $\alpha$ be a root in some field $\mathbb{F}_5^n$. Show that $31$ divides the order of $\alpha \in ...
1
vote
1answer
44 views

Example of a module such that every proper submodule is finitely generated but the module is not.

Let $R$ be a ring with 1 and $M$ an $R$-module. What is an example such that $M$ is infinitely generated but every proper submodule is finitely generated.
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 $k \in \mathbb{N}$, ...
-1
votes
2answers
31 views

every finite integral domain is a field

I am trying to understand a proof that every finite integral domain is a field, and in part is states: "Consider $a, a^2, a^3,\dots$. Since there are only finitely many elements we must have $a^m = ...
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 ...
0
votes
0answers
22 views

Question about valuation rings of a rational function field

Let $k$ be a field and $f \in k[x,y] \setminus k$. Write $E=k(f)$ and $L=k(x,y)$. Assume that there exists a $g \in k(x,y)$ such that $L = E(g)$. Suppose there exists a valution ring $\mathcal{O}$ ...
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 ...
1
vote
1answer
32 views

Ideal quotient and extension

Let $R$ be a commutative ring and $S$ a subring of $R$. If $I$ is an ideal of $S$ define $I^e$ as the ideal in $R$ generated by $I$, i.e. the extension of $I$ in $R$. If $I,J$ are ideals in $S$, we ...
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 ...
0
votes
0answers
28 views

Isomorphic matrix groups over rings

I've thinking about this problem for the last couple days and I can't get anywhere. I would really appreciate some help. Is it true that, a) $\operatorname{SL}_n(\mathbb{Z}/2013\mathbb{Z})\cong ...
0
votes
1answer
26 views

Integral ring extensions and finitely generated as a module

Let $A \subset B \subset C $ be rings. Suppose that $A$ is Noetherian and $C$ is finitely generated as an $A$-algebra. I want to show that $C$ is finitely generated as a $B$-module $ \iff $ $C$ is ...
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) = ...