Questions tagged [ring-homomorphism]

For questions about ring homomorphisms, a function between two rings which respects the structure.

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Such an Integral domain exists or not? [duplicate]

There exists an integral domain $R$ and a surjective homomorphism $R→R$ of rings that is not injective. True or False. This was asked in one of the Ph.D. Selection Exam. I was unable to understand how ...
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k field. What is the correct $n$ and the correct ideal $I\subset k[X_1,\dots ,X_n]$ such that $k[X^2]\simeq k[X_1,\dots ,X_n]/I$ as a $k$-Algebra?

First i thought about $n=1$ and $I=(x-x^2)$, but later i realized that this ideal is the kernel of the homomorphism $$\varphi:k[x]\rightarrow k\times k$$ $$f\mapsto(f(0),f(1))$$ So now i think the ...
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1 vote
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For any ideal, why must a ring homomorphism be surjective for the ideal to be mapped to an ideal?

Problem: Let $f$: $R \to S$ be a ring homomorphism Show that if $f$ is surjective, then for any ideal $I \subset R$, the set $f(I)$ is an ideal of $S$. My Solution: Let $x,y\in I\subset R\implies f(x),...
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The ring $\frac{\mathbb{Z}[x]}{\langle x^n \rangle}$ is isomorphic to?

It is an easy exercise to show that $\frac{\mathbb{Z}[z]}{\langle x \rangle }$ is isomorphic to $\mathbb{Z}$ using the fundamental theorem of ring homomorphisms. I was wondering if the quotient ring $\...
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1 answer
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Does a contradiction occurs about a homomorphism on $C(\Omega)$?

In the page 601 of 'HOMOMORPHISMS OF BANACH ALGEBRAS'(Bade and Curtis) https://www.jstor.org/stable/pdf/2372972.pdf, we get an inequality \begin{align*} ||\nu(x_m - x_n)|| \geq \rho_{\mu(C(\Omega))}(\...
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Prove that if $\Bbb{Q} ≤ E$ and $\Bbb{Q} ≤ F$, then $φ(r) = r$ for all $r ∈\Bbb{Q}$

Suppose $E$ and $F$ are fields, and $φ : F → E$ is a ring homomorphism such that $φ(1) = 1$. I've shown that $φ$ is injective. But if I have $\Bbb{Q} \subset E$ and $\Bbb{Q} \subset F$, how do I show ...
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What is the kernel of this homomorphism?

Let $$q : \mathbb{C}[a_1,a_2,a_3,a_4] \rightarrow \mathbb{C}[u,v]$$ $$a_1\rightarrow u+3v,a_2\rightarrow 3uv+v^2, a_3\rightarrow v^2(3u+v), a_4\rightarrow uv^3$$ be a ring map. My goal is to ...
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Ring isomorphism between a quotient ring and its image under a homomorphism

I've got this question on Atiyah Commutative Algebra (first chapter, "Ideals and Quotient Rings" section). Given a ring $A$ and a ideal $a \subseteq A$, there is a natural projection $\pi: A ...
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Proving that every ring $R$ admits a unique homomorphism $\mathbb{Z} \to R$ [duplicate]

I am trying to prove that there is a unique homomorphism between every ring, R and the integers, $\mathbb{Z}$. I suggested that such a homomorphism $\phi : \mathbb{Z} \to R$ could be defined by for $ ...
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1 answer
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homomorphism from a ring R to the quotient ring R/I

Let $R$ be a commutative ring with $1$ and $I$ be an ideal. There is a natural homomorphism from $R$ to the quotient ring $R/I$ which maps $r$ to $r+I$. Is there any other homomorphism from $R$ to $R/...
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find all ring isomorphism $\phi $of $\mathbb{C}$ onto itself such that for any $a\in \mathbb{R} $ we have $\phi (a)=a$

I am trying to find all ring isomorphism $\phi $of $\mathbb{C}$ onto itself such that for any $a\in \mathbb{R} $ we have $\phi (a)=a$ The isomorphism preserves the algebraic structures $f(x+y)=f(x)+f(...
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Surjective ring homomorphism from $\mathbb{Z}[i] \to \mathbb{Z}_q$

So I've got this question that is asking me to show that the map $\phi: \mathbb{Z}[i] \to \mathbb{Z}_q$ such that $\phi(r + is) = [r] +[s][n]$ is a surjective ring homomorphism where $0 \neq n \in \...
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1 vote
1 answer
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Prove that if X is finite, then X* is a basis for the vector space V* over F.

Here is the full statement of the problem: Let $V$ be a vector space over a field $F$, $X$ be a basis for $V$, and $V$* $=Hom_{F}(V,F)$, which is a vector space over $F$. For each $x\in X$, denote $f_{...
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Homomorphisms between $\Bbb Z[S_3] \to \Bbb Z[\Bbb Z \backslash 2 \Bbb Z]$

Show that there are at most $4$ ring homomorphisms between $\Bbb Z[S_3] \to \Bbb Z[\Bbb Z \backslash 2 \Bbb Z]$. Here is what I did : We know that ring homomorphisms send inversible elements to ...
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Additive inverses in endomorphism ring

I'm working my way through a proof that, given a ring $\langle R, +, \cdot, \theta \rangle$, $\langle \text{End}(R), +, \circ, z \rangle$ is also a ring (where $z(x)=\theta$ for all $x \in R$). ...
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multiplication or composition?

I am studying the ring homomorphism of the following function: If $f: M \to N$ is a smooth function, then $$f^*: C^{\infty}(N) \to C^{\infty}(M), \textbf{ defined by } \phi \mapsto \phi \circ f$$ is ...
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Isomorphism from $\mathbb{Z}[\sqrt{d}]$ to a subring of $M_2(\mathbb{Z})$ [duplicate]

If $d\in \mathbb{Z}$ is square free (we also consider $d=-1$ to be square free), show that the quadratic ring $\mathbb{Z}[\sqrt{d}]$ is isomorphic to the subring $B \subseteq M_2(\mathbb{Z})$ where $$...
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1 vote
1 answer
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Why is a multiplicative inverse preserved in a group homomorphism but lost in a ring homomorphism?

Take $\varphi: R_1 \longrightarrow R_2$, and $f:G_1 \longrightarrow G_2$. In order to prove $\varphi$ is a ring homomorphism, we must show $\varphi(1_{R_1})=1_{R_2}$ (in addition to other properties), ...
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2 votes
1 answer
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Number of injective homomorphisms from complex numbers to quaternions

We are asked to show that there are infinitely many injective ring homomorphisms from the complex numbers to the quaternions $\mathbb{H} =\mathbb{R} + \mathbb{R}i +\mathbb{R}j +\mathbb{R}k$ but I have ...
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continuity of homomorphisms from a unital Banach algebra to a matrix algebra?

Let $A$ be a unital Banach algebra, and let $\phi:A\rightarrow M_n$ be a homomorphism of complex algebras, $M_n$ denoting the algebra of all $n×n$ matrices over $\mathbb{C}$. Is $\phi$ necessarily ...
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3 votes
1 answer
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Ring homomorphism extension from $k[x]$ to $k(x)$

I've been trying to prove that if $k$ is an algebraically closed field, $V$ is a $k$-vector space, $\dim_k(V)$ is infinite, and $f:V\to V$ is a linear operator such that $f-rI$ is invertible for all $...
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4 votes
1 answer
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Is this set a ring?

I am studying Ring Theory for the first time in my life- so the following question may be a very silly one. While trying to solve an (unrelated) exercise, this question clicked in me, and it has been ...
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1 answer
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$\phi: \mathbb{R}[X] \to \mathbb{C}$ is a homomorphism such that: $\phi(X) = 1 + i$. What is $\ker \phi$?

$\phi: \mathbb{R}[X] \to \mathbb{C}$ is a homomorphism such that: $\phi(X) = 1 + i$. What is $\ker \phi$? In my thinking $\ker \phi = \{0\}$ because there is no way to add or multiply $1 + i$ in such ...
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2 votes
2 answers
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Finding the Kernel and Image of $\mathbb Z \to \mathbb Z[i]/(1+3i)$, $x\mapsto x+(1+3i)$

I'm trying to apply the homomorphism theorem to the following function: $$h:\mathbb Z \to \mathbb Z[i]/(1+3i)$$ $$x\mapsto x+(1+3i)$$ Where $(1+3i)$ is the ideal generated by $1+3i$. I know that ...
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0 votes
1 answer
61 views

Homomorphism $\mathbb{Z}[x,y] \rightarrow \mathbb{Z}_{7}$.

I have question to this answer, in homomorphism $\mathbb{Z}[x,y] \rightarrow \mathbb{Z}_7$. I undestand that we have to send $1 \mapsto 1$ to have homomorphism. Author of answer said that if we send $...
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Homomorphisms from $Z[X] $ to $Z_5$. [duplicate]

I would like to know how do homomorphisms from $\mathbb{Z}[X] $ to $\mathbb{Z}_5$ look like. I know that there should be $5$ of them since $1$ must be mapped to $1$ and $\mathbb{Z}_5$ has $5$ elements....
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1 answer
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if $f: \mathbb{R} \to \mathbb{R}$ is a homomorphism and $p$ is an integer-coefficient polynomial, then: $f(p(r))=p(f(r)) \ \forall_{r \in \mathbb{R}}$

Why is that true that if $f: \mathbb{R} \to \mathbb{R}$ is a ring homomorphism and $p$ is an integer-coefficient polynomial, then: $$f(p(r))=p(f(r)) \ \forall_{r \in \mathbb{R}}$$ I know that if $f$ ...
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1 vote
2 answers
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Northcott Multilinear Algebra Universal Property Proof

Northcott Multilinear Algebra poses a problem. Consider R-modules $M_1, \ldots, M_p$, $M$ and $N$. Consider multilinear mapping $$ \psi: M_1 \times \ldots \times M_p \rightarrow N $$ Northcott calls ...
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1 vote
3 answers
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The kernel of the unique homomorphism $\varphi:\mathbb Z\to K$ is a prime ideal.

I am a graduate student of Mathematics.In the book M. Artin Algebra I found a statement: Let $K$ be a finite field.Then the kernel of the unique ring homomorphism from $\mathbb Z$ to $K$ is a prime ...
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2 votes
1 answer
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Show that this function is an isomorphism (tensor product of homomorphisms)

Proposition If $A, B, C$ are modules over a commutative ring $R$, then $Hom_R(A,Hom_R(B,C))\cong_R Hom_R(A\otimes B,C)$. Proof Let $f\in Hom_R(A,Hom_R(B,C))$. We start by showing $$q_f:A\times B\to C, ...
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2 votes
0 answers
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Show that there is one and only one homomorphism such that the tensor product diagram commutes

Proposition Let $M_1,...,M_n$ de modules over a commutative ring $R$. If $(P,p)$ is a tensor product of $M_1,...,M_n$, then, for every multilinear function $q:M_1\times\cdot\cdot\cdot\times M_n\to Q$, ...
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1 vote
1 answer
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Show map between tensor products is a $R-$algebra homomorphism

Prove that, if $A,B,C,D$ are algebras over a commutative ring R, and $f : A \rightarrow B, g : C \rightarrow D$ are homomorphisms of $R-$algebras, then $f \otimes g : A \otimes C \longrightarrow B \...
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1 vote
2 answers
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Prove $\ker \phi=\langle 1+\sqrt{-5}, 2\rangle$

Let $\phi:\mathbb{Z}[\sqrt{-5}]\to\mathbb{Z}_2$ $$\phi(p+q\sqrt{-5})=[p+q]_2$$ I want to prove $\ker \phi=\langle 1+\sqrt{-5}, 2\rangle$ So what I did: $m=p+q\sqrt{5}\in \ker \phi\Longleftrightarrow [...
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1 vote
0 answers
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Prove: $F[t]/H[t]$ is isomorphic to $(F/H)[t]$

Let $F$ a commutative and unitary ring, and also $F$ is an integral domain. Let $H$ be a proper ideal of $F$. I want to prove: $F[t]/H[t]$ is isomorphic to $(F/H)[t]$ So my idea is to use the ...
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0 votes
1 answer
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Let $H=(1+i)\mathbb{Z}[i]$. Let $f:\mathbb{Z}\to \mathbb{Z}[i]/H : f(z)=[z]$. Prove $\ker f=2\mathbb{Z}$.

Let $H=(1+i)\mathbb{Z}[i]$ and let $f:\mathbb{Z}\to \mathbb{Z}[i]/H$ such that $f(z)=[z]$. I want to prove $\ker f=2\mathbb{Z}$. (*) I know $[1+i]=[0] $ So $f(z)=0\Longleftrightarrow [z]=[0]\...
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1 vote
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Ring Homomorphism and Group Homomorphism

I'm new to ring, and I'm currently learning ring homomorphism. From my understanding, it is very similar to group homomorphism as it preserver operation. I wondered if some of the properties (in ...
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Homomorphism given by a multiplication by an integer matrix A

I have the following exercise. Consider a homomorphism given by a multiplication by an integer matrix $A$ given by $\varphi : \mathbb{Z}^k \to \mathbb{Z}^k$. I have to prove that the image of $\varphi$...
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How to prove this conclusion? I don't know if it is related to ring homomorphism.

Assuming $F$ is a number field, and $M_n(F)$ represents the set of all $n\times n$ matrixes on the number field $F$, $M_m(F)$ is defined similarly. Map $f:M_n(F)\to M_m(F)$ meets conditions below: 1) $...
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-1 votes
1 answer
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Under what conditions will the ring homomorphism $\phi : R \to S$ satisfy the following results about prime and maximal ideals?

Let $R$ and $S$ be rings and $\phi : R \to S$ be a ring homomorphism. Here, I am considering that $R$ and $S$ don't necessarily have multiplicative identities. MOTIVATION : I know that the pre-image ...
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0 answers
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Isomorphism of enveloping algebra homomorphisms to $k$-homomorphisms?

Given commutative ring $k$, associative $k$-algebra $A$, and $A$-bimodule $M$, we can define the enveloping algebra $A^e:=A\otimes A^{op}$, where $A^{op}$ is the algebra $A$ but with multiplication ...
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24 views

Why is this homomorphism surjective?

I have the following homomorphism between quotint rings: $$\mathbb{Z}[\sqrt{-5}]/(2) \longrightarrow \mathbb{Z}[\sqrt{-5}]/(2, 1+\sqrt{-5})$$ $(2)$ and $(2, 1+\sqrt{-5})$ denote ideals. I am told that ...
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0 votes
1 answer
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Find all prime ideals in $K[X_1,X_2,X_3]$ which contain $(X_1^2-X_2X_3,\ X_1(1-X_3))$

I have to determine $$\sqrt{(X_1^2-X_2X_3,\ X_1(1-X_3))}$$ in $K[X_1,X_2,X_3]$, where $K$ is a field. I'm supposed to use the fact that $\sqrt{I}=\bigcap_{P\in\text{Spec}A,\ I\subseteq P}P$ for any ...
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4 votes
2 answers
284 views

Prove some properties of ring homomorphism

Let $R,S$ be rings and $\varphi : R\to S$ be a ring homomorphism. Verify that $\varphi(na) = n\varphi(a)$ for all $n\in\mathbb Z$ and $a\in R$. $\varphi(a^n) = (\varphi(a))^n$ for all $n\in\mathbb Z^+...
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1 vote
0 answers
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How to prove something is NOT a ring isomorphism?

I have a question about rings (I assume ring without the requirement of multiplicative identity element). Suppose, I want to show, for two rings, there is NO ring isomorphism (the rings are not ...
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1 vote
1 answer
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Examples of $\mathcal{D}$-chains?

I was reading through this paper and came across an interesting definition: Let $\mathcal{D}$ denote a set of prime ideals of $S$. A chain $\mathcal{C}$ of prime ideals of $R$ is a $\mathcal{D}$-...
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1 answer
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Kernel of ring homomorphism in polynomial ring

Let $K$ be a field with $a_1,...,a_n \in K $ and $\phi : K[x_1,...,x_n] \rightarrow K , \ \phi (f)=f(a_1,...,a_n) .$ I am trying to show that the kernel of this ring homomorphism is the ideal $ I:= (...
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1 vote
0 answers
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A finite ring homomorphism from reals

Is there a finite ring homomorphism $f$ such that $f:\mathbb{R}\rightarrow A$ for some ring $A$? I am not sure where to begin, but I thought perhaps $f:\mathbb{R}\rightarrow \mathbb{R}[x]$ would work ...
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Image of prime and maximal ideals?

Suppose we have a homomorphism $\phi: A \rightarrow B$ where $A$ and $B$ are rings. $\phi$ is injective and not necessarily surjective. Is image of prime ideal and maximal ideals in $A$ also prime and ...
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0 votes
1 answer
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If $f(a)$ is invertible under a ring homomorphism $f$, is $a$ invertible too? [duplicate]

Suppose $f\colon R\to S$ is a ring homomorphism (and not rng homomorphism, wherein $f(1) = 1$ is not generally true). I can prove that if $a$ is invertible$^1$, then $f(a)$ also is, with its inverse ...
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  • 1,967
4 votes
0 answers
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Polynomial of even degree with all odd coefficients has no rational roots

I’d like to show from a ring theoretic approach that any polynomial of odd integer coefficients and even degree has no rational roots. This exercise came from Joseph Gallian’s 9th edition of ...
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