Questions tagged [ring-homomorphism]

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

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Can there be an onto homomorphism from a ring without unity to a ring with unity?

Let $R$ be a ring without a unit element and $R'$ be a (non trivial) ring with a unit element. Can there be an onto homomorphism from $R$ to $R'$? Some observations: There cannot be an isomorphism, ...
Anu's user avatar
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Showing two ring homomorphisms that agree on the integers must agree on the rationals

I have two ring homomorphisms $f,g\colon \mathbb{Q}\to X$. I know that $f=g$ on the integers. How can I show that $f$ and $g$ agree on the rationals? My attempt: let $x,y \in \mathbb{Z}$, $y\neq 0$. ...
Edison's user avatar
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15 votes
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Is there a ring homomorphism $M_2(\mathbb Z)\to \mathbb Z$?

I have the following problem: Is it possible to construct ring homomorphism from $M_2(\mathbb Z)\to \mathbb Z$, or in other words, a homomorphism from ring of all $2\times2$ matrices over the ...
Tino's user avatar
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11 votes
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Can ring homomorphisms be characterized as ring maps such that preimage of any ideal is an ideal?

It's a well know fact that preimage of ideals by ring homomorphism are also ideals. Is the reciprocal true? I.e., let $f:R\to S$ be a map between the rings $R$ and $S$ s.t. $f^{-1}(I)\vartriangleleft ...
Andre Gomes's user avatar
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10 votes
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How do ring homomorphism R → ℤ correspond to prime ideals of R?

In "Category Theory" (Oxford Logic Guides, 2010 by Steve Awodey), pg 35, Awodey makes an off-hand comment: "Ring homomorphisms A → ℤ into the initial ring ℤ ... correspond to so-called prime ideals, ...
Apocalisp's user avatar
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The existence of an algebra homomorphism between $\mathcal{M}_n({\mathbb{K}})$ and $\mathcal{M}_s(\mathbb{K})$ implies $n | s$

Let $n,s \geq 1$ be integers and $\mathbb{K}$ a field. We assume there exist $\Phi : \mathcal{M}_n(\mathbb{K}) \rightarrow \mathcal{M}_s(\mathbb{K})$ an unital algebra homomorphism ($\Phi(I_n)=I_s$). ...
Contestosis's user avatar
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Does there exist a nonzero ring homomorphism from the ring of square rational matrices to the ring of rational numbers?

I am wondering if it is possible to construct a nonzero ring homomorphism from $M_n(\mathbb{Q})$ to $\mathbb{Q}$. So far, I've been unsuccessful in constructing such a nonzero ring homomorphism. Is ...
mayflowers46's user avatar
8 votes
1 answer
1k views

How to "Visualize" Ring Homomorphisms/Isomorphisms?

I understand that the formal definition of a ring homomorphism is some function $f$ that maps $R$ to $S$ ($f: R \rightarrow S$) s.t. $f(ab) = f(a)f(b)$ and $f(a + b) = f(a) + f(b)$. But how do I "...
Max's user avatar
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How would you show that field automorphisms fix prime subfields?

Suppose $K$ is a prime subfield of $E$, then if $\phi$ is an automorphism from $E$ to $E$, we have for all $x \in K$, $\phi(x) = x$. I feel like this is just the definition of a field automorphism, ...
Ninosław Brzostowiecki's user avatar
7 votes
2 answers
576 views

A surjective ring homomorphism which is not an isomorphism [duplicate]

Give an example of a ring $R$ and a surjective homomorphism $R \to R$ that is not an isomorphism. Here is my approach. I know a lemma that says that if $R$ is a Noetherian ring, and $f: R \to R$ is a ...
Albert's user avatar
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7 votes
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Existence of homomorphisms between finite fields

Let $F$ and $E$ be the fields of order $8$ and $32$ respectively. Construct a ring homomorphism $F\to E$ or prove that one cannot exist. Any element $x$ of $F$ satisfies $x^8=x$ and any nonzero ...
user557's user avatar
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If $f(x+y) = f(x) + f(y)$, $f(1) = 1$ and for all $x,y$, $f(xy) = f(x)f(y)$ or $f(xy) = f(y)f(x)$, then $f$ is a homomorphism or an anti-homomorphism

Let $R$ and $R'$ be two rings and $f: R\to R'$ be a map satisfying the conditions: $f(1) = 1$. For all $x,y \in R$, $f(x+y) = f(x) + f(y)$. For all $x,y \in R$, either $f(xy) = f(x)f(y)$ or $f(xy) =...
Cauchy's user avatar
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6 votes
3 answers
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Is restriction of a ring automorphism a subring automorphism?

$\phi:R\to R$ is a ring automorphism. $S \subset R$ is a subring and $\phi(S)\subseteq S$ Is $\phi|_S$ necessarily an automorphism of $S$ ? I can easily check that $\phi|_S:S\to S$ is a homomorphism ...
Maksim's user avatar
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Hom of the direct product of $\mathbb{Z}_{n}$ to the rationals is nonzero.

Why is $\mathrm{Hom}_{\mathbb{Z}}\left(\prod_{n \geq 2}\mathbb{Z}_{n},\mathbb{Q}\right)$ nonzero? Context: This is problem $2.25 (iii)$ of page $69$ Rotman's Introduction to Homological Algebra: ...
user6495's user avatar
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6 votes
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Why does field theory still work with generalized field extensions (embeddings)?

When $E/F$ is a field extension, I've always assumed that $E \supseteq F$. However, there is a more general definition that $E/F$ is a field extension iff $E, F$ are fields and there is an embedding $...
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5 votes
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Existence of ring homomorphism from $\mathbb{Z}[\frac{i}{2}]$ to a finite field

Does a ring homomorphism from $\mathbb{Z}[\frac{i}{2}]$ to a finite field with characteristic $p\equiv 3 \bmod 4$ such that the unity is mapped onto the unity exist? If $F$ is a finite field with ...
kiwi's user avatar
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2 answers
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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 $\mathbb{Z}[x] /\langle x^2 +7 \rangle$ and $\mathbb{Z}[x] /\langle 2x^2 + 7\rangle$ isomorphic?' My sense tells answer ...
Pritam Roy's user avatar
5 votes
2 answers
327 views

Ring endomorphism of $p$-adic integers

I am doing an individual study of an abstract algebra for number theory course online. I just started, so I hope my question just note come off as too trivial. The lecture notes state that the ring of ...
tamboo's user avatar
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2 answers
318 views

Homomorphism of a set to its power set.

Let $(S, +, \cdot, 0)$ and $(S', \oplus, \otimes, 0')$ be two semirings. Then $f: S\rightarrow S'$ is said to be a homomorphism if for all $a, b\in S,$ $f(a+b)=f(a)\oplus f(b)$, $f(a.b)=f(a)\otimes f(...
gete's user avatar
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1 answer
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Prove that there is no homomorphism $f:\mathbb { Z } / n \mathbb { Z }\rightarrow \mathbb { Z }$

Problem Prove that there is no homomorphism $f:\mathbb { Z } / n \mathbb { Z } \rightarrow \mathbb { Z }$ My attempt: By contradiction, let's suppose that there exists as such. We then have $0 = f(0) ...
NotAbelianGroup's user avatar
5 votes
3 answers
550 views

"distributive property" vs. "ring homomorphism": comparing definitions

The property of distributivity is defined using expressions like the following, from page one of "Introduction to Commutative Algebra" by Atiyah and MacDonald: $$x(y + z) = xy + xz$$ $$(y + z)x = yx ...
jessup's user avatar
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On ring isomorphism between cartesian products of rings

Let $R$ be a (non-zero) commutative ring with unity. For $n \in \mathbb N$ denote by $R^n$ the ring under usual co-ordinate wise addition and multiplication. If $R$ is moreover artinian and $R^m \cong ...
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5 votes
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186 views

Counter-example: If $J$ is prime then $f^{-1} (J) $ is prime. $f$ need not be unital.

Let $R$ and $S$ be commutative with 1 and $ $ $ f:R\rightarrow S$ is a ring homomorphism which need not be surjective or unital i.e. $f(1_R)=1_S$ I know that for surjective or unital ring ...
Jhon Doe's user avatar
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Arrow symbol for homomorphism?

Is there a standard arrow or symbol to represent homomorphisms $R\to S$ between two structures? I would prefer not to write out the word homomorphism every time I need to say something is a ...
vkan's user avatar
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1 answer
317 views

Let $A$ and $R$ be rings and $f : A → R$ a ring homomorphism.

Let $A$ and $R$ be rings and $f : A → R$ a ring homomorphism. Give a concrete example of $A, R$ and a ring homomorphism $f : A → R$ such that $A$ is commutative and $R$ is not commutative. Prove that ...
Jellyfish's user avatar
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Let $R$ be a commutative Noetherian ring (with unity), and let $I$ be an ideal of $R$ such that $R/I \cong R$. Then is $I=(0)$?

Let $R$ be a commutative Noetherian ring (with unity), and let $I$ be an ideal of $R$ such that $R/I \cong R$. Then is it true that $I=(0)$ ? I know that a surjective ring endomorphism of a ...
user avatar
5 votes
2 answers
977 views

If a field $F$ is infinite, show that the ring homomorphism $\eta : F[x]\to C(F)$ is one-to-one.

Here is the question I am trying to attempt: I feel like there is a typo... it says let $p(x)=a_{n}x^{n}$. Shouldn't it be $p(x) = a_{n}^{n}\cdots + a_{1}x+a_{0}$? I feel like I must use the roots of ...
user avatar
5 votes
1 answer
142 views

Ring Homomorphism determined by a 6th root of unity

I am working on the following problem : Let $K$ denote the algebraic closure of the 5 element field $F_5$. Let $f: F_5[x] \longrightarrow K$ be the ring homomorphism determined by $f(x) = \omega$, ...
delllaptopswin's user avatar
5 votes
1 answer
140 views

Is the algebraic structure of the full matrix ring preserved by every Lie algebra endomorphism?

Let $A = \mathcal M_n(k)$ be the full matrix algebra over a field $k$. If $\phi:A\to A$ is a nonzero endomorphism of $A$ as a Lie algebra, must it automatically be an endomorphism of $A$ as a unital $...
doetoe's user avatar
  • 3,809
5 votes
0 answers
196 views

No onto map between group algebras $FS_5$ onto $M_6(F)$.

I have to prove that there does not exist a surjective group algebra homomorphism from $FS_5$(the group algebra of the symmetric grpoup, $S_5$, over the field $F$) to $M_6(F)$, where $F$ is the field $...
neelkanth's user avatar
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5 votes
1 answer
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Is this map $\mathbb Q(x, y) \to \mathbb Q((t))$ well-defined?

I am trying to find a map $\mathbb Q(x, y) \to \mathbb Q((t))$ and I have tried the map given by $x \mapsto \sum_{i > 0} t^i$ and $y \mapsto t^{-1}$. I know there is no ring map $\mathbb Q(x, y) \...
algebraist's user avatar
4 votes
3 answers
2k views

What is an Homomorphism/Isomorphism "Saying"?

Outside of the technical definitions, what exactly is a homormorphism or an isomorphism "saying"? For instance, let's we have a group or ring homomorphism $f$, from $A$ to $B$. Does a homomorphism ...
Max's user avatar
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4 votes
1 answer
1k views

What algebraic structure does the set of endomorphisms of a ring have?

Let $R$ be a ring, and let $End(R)$ be the set of ring endomorphisms of $R$, i.e. the set of all ring homomorphisms form $R$ to $R$. Then we can define three binary operations on $End(R)$: $+$, ...
Keshav Srinivasan's user avatar
4 votes
2 answers
135 views

There exist no ring homomorphism $\sigma$ between $\mathbb{Q}_3$ and $\mathbb{Q}_5$ with $\sigma(1)=1$.

There exist no ring homomorphism $\sigma$ between $\mathbb{Q}_3$ and $\mathbb{Q}_5$ with $\sigma(1)=1$. Approach: Since $\sigma(1)=1$, we can conclude that $\sigma(q) = q$ for all $q \in \mathbb{Q}$. ...
bob's user avatar
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4 votes
2 answers
863 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^+...
Owen Murphy's user avatar
4 votes
1 answer
534 views

Counting ring homomorphisms from $\mathbb{Z}[x,y]/(x^3+y^2-1)$ to $\mathbb{Z_7}$

How many ring homomorphisms there is between $\mathbb{Z}[x,y]/(x^3+y^2-1)$ and $\mathbb{Z_7}$? Here $\mathbb{Z_7}$ denote ring of integers mod 7. I don't know how to approach this problem,so far I've ...
2ndYearFreshman's user avatar
4 votes
3 answers
366 views

Example to show that $\mathbb Z^3$ is not a quotient of $\mathbb Z[X]$

I am asked to prove that $\mathbb{Z}^n$ is not a quotient of $\mathbb{Z}[X]$ for any integer $n \geq 3$ by showing that there is no surjective ring homomorphism $\mathbb{Z}[X] \to \mathbb{Z}^n.$ We ...
Jingjie Yang's user avatar
4 votes
1 answer
109 views

How to prove that $\Bbb C[X,Y]/(XY) \ncong \Bbb C[X,Y]/(X) \times \Bbb C[X,Y]/(Y)$

How to prove that $\Bbb C[X,Y]/(XY) \ncong \Bbb C[X,Y]/(X) \times \Bbb C[X,Y]/(Y)$ . I have no idea about this problem on how to proceed, so I couldn't make any attempt.
user avatar
4 votes
1 answer
677 views

Prove that there is a unique homomorphism from $\mathbb{Z} [i]$ to $\mathbb{Z}/2\mathbb{Z}$.

Prove that there is a unique homomorphism from $\mathbb{Z} [i]$ to $\mathbb{Z}/2\mathbb{Z}$. I'm struggling to show uniqueness here. In the past I have shown that $\mathbb{Z}[i]/(1+i)$ is isomorphic ...
jpmacmanus's user avatar
4 votes
2 answers
494 views

$R$ is a subring of $A$ which is a subring of $R[x,y]$ then does there exist an ideal $J$ of $A$ such that $A/J$ and $R$ are ring isomorphic?

Let $R$ be a commutative ring which is a subring of a commutative ring $A$ which in turn is a subring of $R[x,y]$ (all rings and subrings are with unity). Then is it true that there is an ideal $J$ ...
user avatar
4 votes
1 answer
471 views

On an injective ring homomorphism from the ring of continuous functions to the ring of differentiable functions

Let $\phi : C \to D$ be an injective ring homomorphism such that $\phi(1)=1$, where $1$ denotes the constant function $1$ and $C$, $D$ are the rings of continuous, respectively differentiable ...
user avatar
4 votes
1 answer
591 views

Homomorphisms between $\mathbb{Q}(x)$ and $\mathbb{C}$

I want to describe all homomorphisms between the ring of rational $\Bbb{Q}(x)$ functions and the ring of complex numbers $\Bbb{C}$. my idea: Let $\phi:\Bbb{Q}(x)\longrightarrow\Bbb{C}$ be a ...
Gabrielle Rodriguez's user avatar
4 votes
2 answers
2k views

The number of ring homomorphism from $\mathbb Z[x,y]\to \mathbb{F}_2[x]/(1+x+x^2+x^3)$

Find the number of ring homomorphism from $\mathbb Z[x,y]\to \mathbb{F}_2[x]/(1+x+x^2+x^3)$. My attempt: the ring $Z[x,y]$ has three generators $1,x \ and\ y$ we want $1$ to map to $1.$ Since the ...
user345777's user avatar
4 votes
1 answer
3k views

Counting homomorphisms between $\Bbb Z$ and $\Bbb Z_n$ and also between $\Bbb Z_n$ and $\Bbb Q$

Recently, I have tried to find out how many group homomorphisms exist from $\Bbb Z$ to $\Bbb Z_n$ . My argument goes like this: (in the following, $\tau$ and $\phi$ denote the well-known number ...
user avatar
4 votes
1 answer
157 views

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 ...
Sayan Dutta's user avatar
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4 votes
1 answer
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Showing a commutative ring with field subring has a unique ring homomorphism on polynomials [duplicate]

Consider the following problem, given without solution in a german abstract algebra text book: Let $R$ be a commutative ring with $1$ and $K \subseteq R$ a subring, such that $1 \in K$ and $K$ is ...
badapple11's user avatar
4 votes
1 answer
116 views

Are all unital endormorphisms of a Weyl algebra automorphisms?

Given $k$ a field of characteristic $0$, let $A_n(k)$ be the $n$-th Weyl algebra over $k$ – i.e. the unital algebra generated by elements $p_1, ..., p_n$ and $q_1, ..., q_n$ modulo the relations $[p_i,...
Atticus Stonestrom's user avatar
4 votes
2 answers
562 views

If $R_1$ is a division ring, any nontrivial homomorphism $\phi : R_1 \rightarrow R_2$ is injective

For rings $R_1, \: R_2$ let $\phi : R_1 \rightarrow R_2$ be a homomorphism. It's easy to show that $\ker \left (\phi \right)$ is an ideal for $R_1$ Now if we assume $R_1$ is a field (or a division ...
bthmas's user avatar
  • 709
4 votes
1 answer
296 views

Finding ring homomorphisms $\mathbb{Z}\to\mathbb{Z}/n\mathbb{Z}$

A problem from D&F Abstract Algebra: find all ring homomorphisms $\mathbb{Z}\to\mathbb{Z}/30\mathbb{Z}$. Homomorphisms of general rings are assumed (not rings with unity). My reasoning is as ...
lanskey's user avatar
  • 1,066
4 votes
1 answer
74 views

When do contractions respect ideal sums?

Let $\varphi: R \rightarrow S$ be a homomorphism of commutative rings. Given two ideals $I, J \subseteq S$, when does the following equation hold: $$ \varphi^{-1}(I + J) = \varphi^{-1}(I) + \varphi^{-...
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