Questions tagged [ring-homomorphism]

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

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Show that $f$ is a linear mapping and find ker($f$).

Show that $$ f: \left\{ \sum^3_{i=0} a_i t^i \in \mathbb{Z}[t] \right\} \rightarrow \mathbb{Z}^2_3, \quad \sum^n_{i=0} a_i t^i \mapsto ([2]_3a_2 - a_0,a_1 - a_2) $$ is a linear mapping and find ker($f$...
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Automorphism preserves irreducibility?

There is a claim in my notes saying: If $D$ is an integral domain and $f:D[X]\to D[X]$ is an automorphism then $p(x)\in D[x]$ is irreducible iff $f(p(x))$ is irreducible. I have tried to prove this ...
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There exists a unique homomorphism $h \colon K' \rightarrow K$ such that $i = i_k h$

Let $f \colon M \rightarrow N$ be a homomorphism, $K = \ker f$ and $i_k \colon K \rightarrow M$ the inclusion homomorphism. If $i \colon K' \rightarrow M$ is a monomorphism such that $f i = 0$, then ...
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$f\in F[x]$ is separable iff its image under a homomorphism is separable

I am trying to prove that $f\in F[x]$ is a separable polynomial iff $\phi (f) \in K[x]$ is a separable polynomial, where $\phi:F\to K$ is a homomorphism of fields. $f \in F[x]$ is separable iff $f$ ...
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1 answer
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$\operatorname{End_R}(M)$ is an $R$-Algebra if $R$ is commutative

Let $R$ be a ring and $M$ an $R$-module. An $R$-Endomorphism is of $M$ is a $R$-morphism $f: M \rightarrow M$. Now I'd like to show that $\operatorname{End_R}(M)$ with pointwise addition and ...
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2 answers
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Are ring homomorphisms always unital if the rings have $1_{R}$

I know according to some textbooks, rings do not have to contain $1$. But if I define rings to have $1$, are all ring homomorphisms unital? Here's my attempt to prove this: Let $\phi:R\rightarrow S$ ...
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Classification of ring homomorphisms from $\mathbb{Z}[x_{1},\dots,x_{n}]$ to arbitrary unitary ring

I’m working on the following question. Describe all of the ring homeomorphisms from $\mathbb{Z}[x]$ to an arbitrary unitary ring $R$. As I understand this question could be answered with the following ...
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Prove that the map $i$ : R → Q where $i(r) = \frac{rd}{d}$ where d ∈ D is an injective ring homomorphism

Let R be a commutative ring. Let D be any non-empty subset of R that does not contain zero, does not contain any zero-divisors and is closed under multiplication. Then there is a commutative ring Q ...
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Which ideal gives a non trivial quotient ring?

This was an assignment problem and my answer turned out to be correct but I want to know if my concepts are accurate or if I got lucky since this was an MCQ. This question has been asked here and ...
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Kernel of a ring homomorphism

Currently struggling with showing that the quaternions are isomorphic to a subgroup of $M_2(\mathbb{C})$. I've defined a map $\phi$ from $\mathbb{R}\langle x,y,z\rangle$ to $M_2(\mathbb{C})$ such that ...
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How to show that $ \phi$ is Homomorphism .?

I have some confusion on this answer It is written that $F[x,y]/(y^2-x) \cong F[u^2,u] = F[u]$ Or consider $ \phi: F[x,y] \to F[u]$ given by $\phi(x)=u^2, \phi(y)=u$, and prove that $\ker \phi = (y^...
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Examples of an integral extension that $ht P \lt ht P\cap A$

All rings are commutative Noetherian with 1. Matsumura's Exercise 9.8. Let $A$ be a ring and $A\subset B$ an integral extension. If $P$ is a prime ideal of $B $ with $p = P\cap A$ then $ht P \leq ht p$...
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Ring homomorphisms from $C[0,1] \to \Bbb R$ all of certain form [duplicate]

Let $R$ be the ring of real-valued continuous functions on $[0,1].$ Note that if $0 \le t \le 1$ then the evaluation map $\psi_t :f \mapsto f(t)$ is a homomorphism of $R$ into $\Bbb R$. Show that any ...
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The ideal $\langle x-y+1,y-3\rangle$ of $\mathbb C[x,y]$ is maximal

I'm trying to show that the ideal $I =\langle x-y+1,y-3\rangle$ of the ring $\mathbb C[x,y]$ is maximal. My approach so far: We define $\phi:\mathbb C[x,y]\to \mathbb C,$ with mapping $$f(x,y)\...
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Showing two given rings are isomorphic

Let $K$ be a field and $R=K[x,y]$ be a polynomial ring in the variable $x$ and $y$. let $R_1=R[y/x]$ is a subring of the quotient field of $R$. Let $R_2=R[t]/(xt-y)$. show that $R_1$ and $R_2$ are ...
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Let $R$ be a ring. Find the pairs $(a,b) \in R \times R$ such that $f: R \to R$ defined by $f(x)=axb$ is a ring homomorphism.

Let $R$ be a ring. Find the pairs $(a,b) \in R \times R$ such that $f: R \to R$ defined by $f(x)=axb$ is a ring homomorphism. What I want is that for $x,y \in R$ $$\begin{align*}f(x+y)&=f(x)+f(...
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Let $R$ and $S$ be rings of characteristic $0$. If $R$ and $S$ are isomorphic as a ring, then, why $R/7R$ is isomorphic to $S/7S$ as a ring?

Let $R$ and $S$ be rings of characteristic $0$. If $R$ and $S$ are isomorphic as a ring, then, why $R/7R$ is isomorphic to $S/7S$ as a ring ? My try : Let $φ:R→S$ be an isom and $π: S→S/7S$ be ...
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Number of unit preserving ring homomorphism

Recently a few weeks back I was faced with an admission test question. The question goes as follows: The number of unit preserving ring homomorphisms from the ring $\mathbb{Z}[\sqrt{2}]$ to $\mathbb{...
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Kernel of ring homomorphisms and subring test

Let $R$ be a ring and we adjoin an element "$a$" to the ring $R$ with some relation $f(a) = 0$. The resulting ring is $R[x]/\langle f\rangle= R'$ (say). Now latest consider the inclusion map ...
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1 answer
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Finding kernel of a ring homomorphism [closed]

I am reading Algebra by Artin and facing a problem in the following example: Let f : R[x,y] to R[t] be a ring homomorphism that sends x to t² and y to t³ i.e. it sends g(x,y) to g(t²,t³). Then the ...
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Isomorphism between two quotient ring [closed]

I am self studying Artin's algebra and stuck with this exercise which asks 'Are $Z[x] /\langle x^2 +7 \rangle$ and $Z[x] /\langle 2x^2 + 7\rangle$ isomorphic?' My sense tells answer is no but to show ...
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An elegant closure property of uniform distributions?

In dice notation, any collection of polyhedral dice can be represented as a polynomial in one unknown. For example, $1d^{20} + 3d^6$ denotes a collection consisting of one 20-sided die and three 6-...
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Prove that $\phi(p(x)) = p(a)$ is a ring isomorphism from the polynomial ring $\mathbb{F}(x)$ to the ring $\mathbb{F}$

I have a proof for this, it was quite easy and straightforward, but that's the problem. It seemed too easy, I just want to make sure I didn't miss any important nuances or simplify things too greatly. ...
2 votes
1 answer
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Show that two rings of sequences are non-homomorphic

Let $A=(a_0, a_1, a_2, a_3,...)$ and $B=(b_0, b_1, b_2, b_3,...)$ denote infinite integer sequences. Define addition coordinate-wise as $A+B=(a_0+b_0, a_1+b_1,a_2+b_2,a_3+b_3,...a_n+b_n,...)$ and ...
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No. of ideals of a ring R if cardinality of set of all homomorphic images (up to isomorphism of R) is n.

i do not understand how to start the question. what is homomorphic images (up to isomorphism)?
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Injective ring homomorphism $f: \Bbb Z/ n\Bbb Z \to A$ does not imply $A$ is an integral domain

Let $A$ be a ring such that there exists an injective ring homomorphism $f: \Bbb Z/ n\Bbb Z \to A$ for $n \geq 2$. I know that if $A$ is an integral domain, then $n=p$ where p is prime but is the ...
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Ring homomorphism from $\Bbb R \to \Bbb C$ must be inclusion? [duplicate]

I was wondering what are the ring homomorphism $f$ from $\Bbb R \to \Bbb C$ and to $\Bbb C \to \Bbb C$ ? I think that there is only one ring homomorphism from $\Bbb R \to \Bbb C$ : the inclusion. I ...
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Prove that the kernel of a ring homomorphism is an ideal.

The problem statement is as follows: An ideal is a subring $\mathcal{J} \subset R$ with the property that $\forall r \in R\ \text{and}\ a \in \mathcal{J}$ we have $$ra,ar \in \mathcal{J}$$Prove that ...
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Let $\varphi : R \to R'$ be a surjective ring homomorphism. Let $I \subset R'$ be a maximal ideal. Then show that $\varphi^{-1}(I)$ is maximal ideal.

Let $\varphi : R \to R'$ be a surjective ring homomorphism. Let $I \subset R'$ be a maximal ideal. Then show that $\varphi^{-1}(I)$ is maximal ideal. This question already has answer, but I couldn't ...
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Prove $U^{-1}R \cong R[\{X_s\}_{s \in U}]/(sX_s - 1).$

Let $I = \langle \{sX_s - 1 : s \in U\}\rangle.$ I'm looking for an alternate proof of $U^{-1}R \cong R[\{X_s\}_{s \in U}]/I.$ The proof I found is sending $r/u \to rX_u,$ proving this is a well-...
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How can I understand this construction of a surjective $R$-module homomorphism?

I have the following construction A system of generators $(x_i)_{i\in I}$ of an $R$-module $M$ induces a surjective $R$-module homomorphism $$\bigoplus_{i\in I}R\rightarrow M,~~e_i\mapsto x_i$$ I ...
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$f:R\to S$ is finite injective and $S$ is of dim $1$

Let $R$ be a local noetherian domain of dim $3$. I'm asked to construct examples $f:R\to S$ with the following properties: 1- $f$ is onto and $S$ is of dim $1$. 2- $f$ is injective and $S$ is of dim $...
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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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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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1 answer
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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 \...
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_{...
1 vote
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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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2 votes
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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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1 answer
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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 $$...
1 vote
1 answer
106 views

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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