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
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Characterizing kernel of ring morphism

Let $K$ be a field and define a ring morphism $\psi: K[x_1,x_2, \dots , x_n, y_1, y_2, \dots , y_n] \rightarrow K(x_1,x_2, \dots , x_n)$ by $\psi(x_i) =x_i$ and $\psi(y_i) =\frac{1}{x_i}$. I think ...
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0answers
22 views

Commutative semiring such that every element except zero does not have additive inverse and each element can be uniquely sum-decomposed

Is there a unique factorization commutative semiring such that every element except zero does not have additive inverse in the semiring and each element can be decomposed into unique finite sum ...
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2answers
52 views

Show that there is no surjective ring homomorphism from $\mathbb Z_2[x]$ to $\mathbb Z_2 \times \mathbb Z_2\times \mathbb Z_2$

I saw this question as a bonus from a past exam, and here's my solution for verification. I argued like so. I said suppose there is such a surjective homomorphism $f$, then $f(0)=(0,0,0)$, $f(1)= ...
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1answer
52 views

What are some examples of coolrings that cannot be expressed in the form $R[X]$?

Let $K$ denote a field. Then the polynomial ring $K[x]$ has the property that the sum of two units is either a unit, or zero. I'll bet there's heaps of other examples, though. So let a coolring be a ...
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1answer
79 views

Paradox in ring theory — what am I missing?

I saw somewhere the following exercise: Give example of prime ideal in a ring which is not maximal the answer was this: Let $R$ be our Ring and $I$ ideal such $$ R = {Z}[{X}] $$ $$ I = (x) $$ ...
2
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1answer
75 views

Is there a theory of “rings” with partially defined multiplication?

Consider the abelian group $R [[\mathbb{Z}^d]]$ of all formal Laurent series over a commutative ring $R$ (a typical element has the form $\sum_{v \in \mathbb{Z}^d} \lambda_v \cdot X_1^{v_1} \dots ...
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0answers
25 views

Is there some efficient way besides Eisentein's criterion to show that polynomials are irreducible?

While solving some problems, I had to show that $y-x^2$, $y^2-x^3$ are irreducible in $k[x,y]$ ($k$ is an algebraically closed field). The Eisenstein criterion don't apply here. Is there some other ...
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0answers
44 views

Symmetric Algebra and Extension of Scalars

I am reading Gortz and Wedhorn's Algebraic Geometry. In their section on the symmetric algebra they explain the adjoint situation $$ \mathrm{Sym}_A \dashv i_A\colon \mathrm{Alg}(A)\to ...
3
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1answer
89 views

Prove that the ring of rational numbers $\Bbb Q$ is not isomorphic to the ring of real numbers $\Bbb R$

just wondering if my reasoning is correct. I said assume there is such a homomorphism f, then f(1)=1 since it is a ring homomorphism. But $$f(\sqrt 2)= f(1\cdot\sqrt 2)= f(1) \cdot \sqrt 2= \sqrt ...
6
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1answer
48 views

The number of combinations $(a,b) \in \mathbb{Z}_n \times \mathbb{Z}_n$ such that $a \cdot b = 0$

This question is about a ring for some chosen $n \in \mathbb{N}$ I wanted to find the number $M_n$ of combinations $(a,b) \in \mathbb{Z}_n \times \mathbb{Z}_n$ can be found such that $a \cdot b = 0$ ...
1
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1answer
34 views

Failure of the Krull-Schmidt Theorem?

Theorem 1.19 of Representation Theory of Finite Groups: Algebra and Arithmetic is the Krull-Schmidt theorem, which I screenshotted and uploaded it here, I don't have any problem with this theorem and ...
2
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3answers
66 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
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1answer
38 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,/] ...
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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 ...
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0answers
27 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 ...
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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 ...
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0answers
11 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 ...
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0answers
36 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 ...
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1answer
49 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
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1answer
65 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 ...
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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) := ...
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3answers
106 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
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1answer
22 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 ...
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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
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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 ...
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3answers
65 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 ...
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3answers
95 views

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

I am trying to factor $X^{20}-1$ 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 $20X^{19}$ ...
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0answers
23 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
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0answers
37 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 ...
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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
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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 ...
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4answers
147 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
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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 ...
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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 ...
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0answers
29 views

How I can find the Jacobson radical? [closed]

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.
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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
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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 ...
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2answers
62 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 ...
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0answers
38 views

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

Let $R$ be a commutative ring with unity. Show that if $I_M=J_M$ for all $M \in\operatorname{Max}(R)$ then $I=J$.
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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 $L=K[\alpha]$. ...
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0answers
61 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 ...
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0answers
40 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 ...
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0answers
33 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?
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0answers
59 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.
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0answers
20 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
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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.
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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 ...
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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 $k \in \mathbb{N}$, ...
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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 = ...