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Questions tagged [ring-homomorphism]

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

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Proof verification: Is the quotient ring $\Bbb C[x]/(x^2+1)$ an integral domain? [duplicate]

Is the quotient ring $\Bbb C[x]/(x^2+1)$ an integral domain? My solution goes like this: If possible let us assume that $\Bbb C[x]/(x^2+1)$ an integral domain. This means $(x^2+1)$ is a prime ideal in ...
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Ring homomorphism from a ring to a field

Since Any ring homomorphism must map group identity to group identity, but on the left hand side is a field and it has two group identities. So if I consider a ring $\mathbb{Z}_6$ and a field $\mathbb{...
Maths wizard's user avatar
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LAnn$_R(a)$ denotes the kernel of some ring homomorphism.

Let LAnn$_R(a)$ denote the collection of all left annihilators for arbitrary element $a\in R$. I'm curious to show that this subset LAnn$_R(a)\subset R$ denotes an ideal specifically by finding a ring ...
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How many homomorphism are from Z to the ring of matrices?

Let be $(\mathbb{Z},+,\cdot)$ and $(\mathcal{M}_{2x2}(\mathbb{Z}) , + ,\cdot)$ rings, and $\phi: \mathbb{Z} \longrightarrow \mathcal{M}_{2x2}(\mathbb{Z})$ a function. How many ring homomorphisms $\phi$...
pucky's user avatar
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doubt about ring homomorphism. in $f(a)=a^2$

Consider the following statements (a) If R is a commutative ring with unity and $f:R\to R$ be a ring homomorphism defined by $f(a)=a^2$ then $1+1=0$ (b) If R is a commutative ring with unity and $f:R\...
math student's user avatar
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All module homomorphisms from $\mathbb{Z}^n$ to $\mathbb{Z}$

This is a question on something on something more general, but for now I'd like to keep in simple. Consider a module homomorphism $\phi:\mathbb{Z}^n\to\mathbb{Z}$, where $n$ is a positive integer. ...
Num2's user avatar
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A question about the embedding from $\mathbb{Q}(\sqrt{2},\sqrt{3})$ to an algebraic closure of $\mathbb{Q}$

I am now just beginning my study in field theory and I am trying to find all embeddings from $\mathbb{Q}(\sqrt{2},\sqrt{3})$ to $\bar{\mathbb{Q}}$ (an algebraic closure of $\mathbb{Q}$). Here, an ...
ZYX's user avatar
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2 votes
1 answer
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Isomorphism between two fields can be extended to isomorphism between their respective closures

Let $K_{1}$ and $K_{2}$ be two isomorphic fields. Prove that an isomorphism $f \colon K_{1} \xrightarrow{\sim} K_{2}$ can be extended to an isomorphism $\sigma \colon \overline{K_{1}} \xrightarrow{\...
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existence of a ring isomorphism from A to the ring of complex numbers

Consider the $C_3$-representation $\rho$ on $\mathbb{R}^2$ that sends a generator of $C_3$ to the counter-clockwise rotation of the plane by $120$ degrees. Consider the morphisms $A = Hom_{C_3}(\rho,\...
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What are the module homomorphisms from $\mathbb Q$ to $\mathbb Z/2\mathbb Z$? [closed]

What are the module homomorphisms from $\mathbb Q$ to $\mathbb Z/2\mathbb Z$, looking at $\mathbb Q$ as a $\mathbb Z$-module? The impression I got from the proof in the book is that it is the zero ...
Intuition's user avatar
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homomorphisms from a (semi)local ring to a ring with infinitely many maximal ideals

In continuation of the this question: homomorphism from a (semi)local ring to $\mathbb Z$. I tried to construct (unital) homomorphisms from a (semi)local ring to a ring with infinitely many maximal ...
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Check finiteness of ring map with SAGE

(I asked this question in a SAGE-specialized forum --see here--, but did not received an answer there sofar. I therefore decided to post the question also here.) Let $R \rightarrow S$ be a ring map. I ...
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homomorphism from a (semi)local ring to $\mathbb Z$

One can easily construct homomorphisms from $\mathbb Z$ to a (semi)local ring $\mathbb Z/6\mathbb Z$, even a field, $\mathbb Q$. How about the converse? Is there a homomorphism from a (semi)local ...
user1401's user avatar
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Is the following transformation h from R to S a ring homomorphism?

Do the following ring homomorphisms with identity exist from $R$ to $S$? a) $R=\mathbb{Q}[X]$, $S=\mathbb{Q}$, $f(X^2-2)=0$ b) $R=\mathbb{R}[X]$, $S=\mathbb{C}$, $f(X^2+4)=0$ I know the following ...
Jimmie Hilberto's user avatar
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On the homomorphism as $M_n(A)$-modules.

I'm trying to prove that, if $A$ is an Algebra over a field $F$, and $U,V$ are $A-$modules. Then any element in $Hom_{M_n(A)}(U^n, V^n)$ (recall that $U^n, V^n$ is the set of size $n$ vectors as an $...
Nestor Bravo's user avatar
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Show that: exists unique ring homomorphism $\varphi : \mathbb{Z} [ i ] \longrightarrow R$ with $\varphi ( i ) = a$, where $a^{2} =-1_{R}$

Problem: Let R be a ring which has an element $a \in R$ such that $a^{2}=-1_{R}$. Prove that: there exists a unique ring homomorphism $\varphi : \mathbb{Z} [ i ] \longrightarrow R$ such that $\varphi (...
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Let $R$ and $R'$ be two rings and $f:R\to R'$ be a ring homomorphism. Let $I$ be an ideal of $R.$ Is $f(I)$ an ideal of $R'?$ Justify. [duplicate]

Let $R$ and $R'$ be two rings and $f:R\to R'$ be a ring homomorphism. Let $I$ be an ideal of $R.$ Is $f(I)$ an ideal of $R'?$ Justify. I can prove that $f(I)$ is a subgroup of $R'.$ This is because, ...
Thomas Finley's user avatar
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Relation of ring homomorphism pre-images

Currently I'm working on my final project, and while I read some literature about my project, I found this expressions: Given $f: R\to R'$ is a ring homomorphism. For any $y_1,y_2\in R'$, $$ \{x \mid ...
ausfear's user avatar
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Let $K_1, K_2$ be fields and $f: K_1 \to K_2$ Ring Homomorphism. Prove $f$ is injective or Zero Homomorphism ($f(a) = 0_{K_2}$ for all $a \in K_1$)

By the way, is "Zero Homomorphism" the correct translation of the German word "Nullhomomorphismus" ? When I googled, I did not find the word Null Homomorphism in English Let $K_1$ ...
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Equivalent statement about the characteristic of a ring

I am currently self-learning ring theory and I stumbled over two equivalent statements describing the characteristic of a ring R. Given a commutative ring R, the characteristic of R is the smallest ...
196884 is 196883 plus 1's user avatar
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Commutativity of a ring with a ring homomorphism to itself $f(x) = x^2$

The full question is about a bit more than just commutativity, but I'm stuck on the commutativity part right now: Let $R$ be a ring with the property that $f : R \rightarrow R, f(x) = x^2$, is a ring ...
Bebedaeh's user avatar
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Surjective ring homomorphism from $\mathbb{Z}[\sqrt{-5}]$ to $\mathbb Z/3\mathbb Z$. [closed]

Question: Let $R=\mathbb{Z}[\sqrt{-5}]$ and let $$ \phi: \mathbb{Z}[\sqrt{-5}] \rightarrow \mathbb{Z} / 3 \mathbb{Z}, \quad a+b \sqrt{-5} \mapsto \overline{a+b} \quad(a, b \in \mathbb{Z}) . $$ (a) ...
illegalsh's user avatar
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1 answer
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What $\mathbb Z/\ker f$ mean if $f$ is injective?

Here, I am trying to really understand the concept of cosets. From my understanding, given groups A and B, then A/B would mean the set of all the cosets of B in A. In that case, if I have $\mathbb Z/\...
Mr Prof's user avatar
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$ \left\lfloor\frac{x - \pi(z)^*}{n} \right\rfloor\geq\left\lfloor\frac{x-z^*}{m}\right\rfloor $ whenever $n\mid m$ and $\pi:\Bbb{Z}/m\to\Bbb{Z}/n$✨

If $z \in \Bbb{Z}/n$, w let $z^* =$ the standard residue or in other words the least non-negative integer equal to $z$ modulo $n$. Suppose that $n \mid m$ for some two positive integers $n,m$. If $\pi ...
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Small question about contravariant functors

I am new to Category Theory and it was recommended to me to start my understanding of basic concepts by reading $\textit{Basic Category Theory}$ by Tom Leinster. But I am finding myself struggling a ...
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Does there exist a surjective but not injective homomorphism of rings from an integral domain R → R

I could come up with examples for rings;say for example defining the homomorphism to be a left shift of elements which are taken to be of the form of a countably infinite tuple.Can someone help me out ...
Bavanesh B S's user avatar
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Impact of plaintext modulus overflow in (B)FV homomorphic encryption

I just tried reading the paper "Somewhat Practical Fully Homomorphic Encryption" by Fan and Vercauteren but I got confused by their use of the coefficient-wise modulus. They define the ...
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1 vote
2 answers
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Number of homomorphisms from $Q[x]/\langle f(x)\rangle$ to $\mathbb{C}$

How many homomorphisms are there from $Q[x]/\langle f(x)\rangle$ to $\mathbb{C}$ that take $1$ to $1$ for an arbitrary polynomial $f(x)\in Q[x]$? I took some examples and tried to figure out the ...
nkh99's user avatar
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2 answers
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Is the kernel of a ring homomorphism a subring or an ideal?

Is the kernel of a ring homomorphism a subring or an ideal? Dummit & Foote (Abstract Algebra, $3^{rd}$ ed., 2004.) state in Proposition 7.3.5 (2), page 239, Proposition 5. Let $R$ and $S$ be ...
Cliff's user avatar
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2 votes
1 answer
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Why are the additive idenity elements of two rings assumed to be the same in the Lemma?

I encountered the following lemma about homomorphisms in Ring Theory. I found the lemma stated in the book, "Topics in Algebra " by I.N Herstein (on Chapter-2, Ring Theory, Page no-131, 2nd ...
Thomas Finley's user avatar
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Question on passage in a tensor product proof

I was studying this proposition: Let $A$ be a finite dimensional unital $F$-algebra, let $K/F$ be a field extension and $n\geq 1$. Then: $M_n(F)\otimes_F A \simeq M_n(A)$ is a $F$-algebra isomorphism....
James Cats's user avatar
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What does "factoring" mean in the context of ring epimorphisms?

Let $f\colon A \to B$, $g\colon A \to C$ be two ring epimorphisms. What does it mean when one says that these two factor into another ring epimorphism $h\colon B \to C$? I'm guessing that it means ...
NoetherNerd's user avatar
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1 answer
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Applying ring homomorphism to coefficients of polynomial is a ring homomorphism? [closed]

Let $R_1$ and $R_2$ be rings, $\phi:R_1\to R_2$ some ring homomorphism. Consider the map $\widehat\phi$ that sends any polynomial $f(x) = \sum_ia_ix^i\in R_1[x]$ to $\widehat\phi(f(x)) = \sum_i\phi(...
node196884's user avatar
1 vote
1 answer
104 views

What is the set $S$ of rings $R$ such that, for all rings $R'$, there is at most one nonzero homomorphism $R \to R'$?

What is the set $S$ of rings $R$ such that, for all rings $R'$, there is at most one nonzero homomorphism $R \to R'$? We're dealing only with commutative rings with unity, and our definition of ring ...
shintuku's user avatar
1 vote
2 answers
122 views

Isomorphism between $R/P$ and $R_P/PR_P$

Let R be a commutative ring, and P a prime bilateral ideal. Let $R_P = \left\{ \frac{a}{s} \ \middle\vert \ a \in R, \ s \not \in P \right\}$ and $PR_P = \left\{ \frac{a}{s} \ \middle\vert \ a \in P, \...
Rararat'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
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Define $\varphi_a:F\to\Bbb R$ by $\varphi_a(f)=f(a)$ for $f\in F$, then $f:\Bbb R\to\Bbb R$ is a ring homomorphism for each $a\in\Bbb R$.

Below is a proof I've written to show a ring homomorphism. This is how I was taught to write them, with the proposition, then the proof. So if it's redundant, I'm sorry. What I'm asking is whether or ...
Tacosi's user avatar
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2 votes
2 answers
142 views

Show $f^{-1}(I)$ Ideal in $R$? [closed]

I'm trying to show that if $f: R \rightarrow S$ is a ring homomorphism, and $I \subset S$ is ideal, then $f^{-1}(I)$ is Ideal in $R$ I'm trying to show that is satisfies the conditions of an ideal, ...
freestyle4dayz's user avatar
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2 answers
80 views

How do you prove a ring homomorphism is onto using the range and kernel?

The problem is: Suppose $A$ is a commutative ring and $a \in A$. If $a^2 = a$, prove that the function $\pi_a(x) = ax$ is a homomorphism from $A$ into $A$. Show that the kernel of $\pi_a$ is $I_a$, ...
dryoung's user avatar
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3 votes
0 answers
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Ring homomorphism from ring to subring that fixes subring

Let $A \le B$ be rings. Suppose that there exists a unique ring homomorphism $ \phi : B \rightarrow A$ such that $ \phi (a) = a$ for all $a \in A$. Does it follow that $A=B$? I proved that if $B$ is ...
user1142333's user avatar
2 votes
1 answer
96 views

When are endomorphism rings Dedekind-finite?

The Question Is it possible to characterise the abelian groups $G$ whose endomorphism rings are Dedekind-finite? Or, at least, are there conditions which are necessary/sufficient for this to occur? (A ...
Joe's user avatar
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1 vote
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Simplified proof of Chevalley's theorem for $A\to k$. [duplicate]

I am looking for a simplier proof for the following special case of Chevalley's theorem: Theorem. If $A\subset k$ is a subring of a field and the inclusion $i: A\to k$ is finitely presented, then $\{(...
William Sun's user avatar
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0 votes
1 answer
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When is a quotient $\mathbb Z$-algebra a ring quotient

Suppose that I have two rings with identity, $S,R \in $ Ring, and a surjective $\mathbb Z$-algebra homomorphism $f: S \to R$ (hence $R$ is a quotient $\mathbb Z$-algebra of $S$). Is it true that $R$ ...
Adelhart's user avatar
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1 vote
0 answers
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Onto map between the group algebras.

Let $\mathbb{F}_p$ be a finite field of order $p$ ($p$ prime) and $G=SL(2,7)$ be a special linear group of $2\times 2$ matrices over the field $\mathbb{F_7}.$ Let $\mathbb{F}_pG$ be the group ring (or ...
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1 vote
1 answer
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Augmentation map when ring is not commutative

I am reading a book in representation theory and it says for any group $G$ and a commutative ring $R$ with a $1$. Then the map $$ \epsilon : RG \rightarrow R \\ , g \mapsto 1 \text{ for all } g \in ...
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Endomorphisms between direct product/sum of algebraic objects?

My motivation is this Wikipedia article. Suppose $R$ be a ring with unity and $M,N$ be $R$-modules. Take their direct sum/product $P=M \oplus N$. So $P$ is also a $R$-module. Consider the respective ...
MAS's user avatar
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Definition of the kernel of a ring homomorphism with 1

I came across this problem: Let $R$ be a integral domain. Show that the map $b \mapsto ba$, with $a \neq 0$, is injective. My proof went as follows: Since $R$ is an integral domain, we have $ab = 0 \...
Kajice's user avatar
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2 votes
2 answers
157 views

examples of ring homomorphism $f : R → S$, where $R$ is arbitrary

I've read the definition of the ring homomorphism: Definition. Let $R$ and $S$ be rings. A ring homomorphism is a function $f : R → S$ such that: (a) For all $x, y ∈ R, f(x + y) = f(x) + f(y).$ (b) ...
user1401's user avatar
2 votes
2 answers
125 views

Number of ring homomorphisms

I'm trying to solve the following problem: Determine how many different ring homomorphism from $\mathbb{Z}[i] \to \mathbb{Z}/(85)$ exist. For a previous question I had to determine the unique integer ...
hizerain's user avatar
2 votes
1 answer
173 views

Injective but not surjective R-module homomorphism

Let $R$ be a PID. I want to find an injective but not surjective $R$-module homomorphism $\varphi:R^n\to R^n$ for some $n\geq 1$. I can find injective but not surjective ring homomorphism (for $n=1$). ...
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