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Questions tagged [ring-homomorphism]

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

2
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1answer
23 views

Find a homomorphism that maps a point in a Boolean algebra into the image of a proper filter

Let $\mathbb B$ be a Boolean algebra. Let $F$ be a proper filter in $\mathbb B$ (i.e. $0 \notin F$), and let $I$ be its dual ideal. Suppose there exists $a \in \mathbb B$ such that $a \notin F \cup I$....
3
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2answers
29 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 ...
3
votes
1answer
35 views

Showing this is an automorphism

Let $R_n = F_q[x]/\langle x^n - 1 \rangle$, where $F_q[x]$ is a finite field. Consider $\mu_a$ which acts on $R_n$ like so; $f(x) \mu_a \equiv f(x^a) \bmod (x^n - 1)$ for $f(x) \in R_n$. Is this an ...
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0answers
20 views

Example of a Jordan homomorphism that is not a homomorphism or antihomomorphism

Can anyone please provide an example of a Jordan homomorphism (preferably on $n\times n$ matrices over a commutative ring) that is not already a homomorphism or antihomomorphism? An obvious Jordan ...
1
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1answer
51 views

Ring Homomorphism Counterexample

Suppose you have the following conditons: $f(x+y) = f(x) + f(y), f(xy) = f(x)f(y)$ and $f(1) = 0$, for all $x, y$. Would these conditions be sufficient to form a ring homomorphism always or is there a ...
2
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1answer
59 views

Finding a unique ring homomorphism

Let $R$ be a commutative ring. I have to show that for any $a,b\in R$, there exists a unique ring homomorphism $f:R[X]\to R[X]$ such that $f(c)=c$ for all $c\in R$ and $f(X)=aX+b$. I am not getting ...
3
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1answer
24 views

What elements does a map of rings fix if both rings contain the same algebraically closed field as a subring?

Let $k$ be an algebraically closed field. Let $R,S$ be rings such that $R,S$ both contain $k$ as a subring. Let $\varphi:R\to S$ be a ring homomorphism. Then does $\varphi(a)=a$ for all $a\in k$? So ...
4
votes
1answer
44 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 $...
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2answers
58 views

Questions about the function $f:\Bbb Z_{8}\rightarrow \Bbb Z_4$

I have the function $f:\Bbb Z_{8}\rightarrow \Bbb Z_4$ without any particular expression associated. How many surjective functions $f:\Bbb Z_{8}\rightarrow \Bbb Z_4$? How many of them are a ...
3
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3answers
36 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 ...
1
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1answer
18 views

Constant Endomorphism

Is the constant map $f$:M $\longrightarrow$ M with m $\longmapsto$ a is an endomorphism , where M is a module? let $m ,m' \in M,$ we have $f(m)=f(m')=a $ then $f(m)+f(m')=2a $ but $f(m+m')=a$ ...
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3answers
117 views

Some questions about $f: \Bbb Z_{44100} \rightarrow \Bbb Z_{150}\times\Bbb Z_{294}$

Given the function: $f: \Bbb Z_{44100} \rightarrow \Bbb Z_{150}\times\Bbb Z_{294}$ defined as follows $[x]_{44100} \rightarrow ([x]_{150},[x]_{294})$ Calculate $f(12345)$ - Answered A preimage of (...
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1answer
45 views

Cayley's Theorem for Composition Rings?

A composition ring is a commutative ring $(R,+,\cdot)$ endowed with an additional binary operation $\circ$ satisfying the following properties for all $f,g,h\in R$: $(f+g)\circ h=f\circ h +g\circ h$ $...
3
votes
1answer
298 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 $End(R)$: $+$, ...
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0answers
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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 $...
1
vote
1answer
54 views

Doubt about ring homomorphism.

My question is related to this question. For convenience I'm giving the question : How many ring homomorphisms there is between $\mathbb{Z}[x,y]/(x^3+y^2-1)$ and $\mathbb{Z_7}$? Here $\mathbb{Z_7}$ ...
0
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1answer
75 views

Why is $f(x) = 5x$ not a homomorphism?

Why is the function $f: \mathbb Z \to \mathbb Z$ given by $f(x) = 5x$ not a homorphism, since $f(a+b) = 5(a+b) = 5a + 5b = f(a) + f(b)$, and same for $f(a*b)$.
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2answers
40 views

Show $f:\mathbb{Z}_6 \rightarrow \mathbb{Z}_3$ is a homomorphism

I am currently studying for an abstract algebra final exam. I am trying to disprove the statement "Consider the homomorphism $f: R \rightarrow S$ where R and S are rings. Prove/Disprove: If $a \in R$ ...
2
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2answers
60 views

Injective Homomorphism from $\mathbb{R}\times\mathbb{R}$ to the ring of Continuous functions

Does there exist an injective ring homomorphism from the ring $\mathbb{R}\times\mathbb{R}$ to the ring of continuous functions over $\mathbb{R}$? I know that $\mathbb{R}\times\mathbb{R}$ is a field. ...
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2answers
22 views

Algebra - Homomorphism

When checking if two rings are isomorphic, we check if mapping is homomorphism and then we check if it is bijective (injective and surjective). In some tasks when checking if isomorphisms, we checked ...
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1answer
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Algebra - endomorphisms of field

Find all endomorphisms of $\mathbb{Q}$. ($\mathbb{Q}$ is the field.) When finding isomorphism of $2\mathbb{Z}$ and $3\mathbb{Z}$, we define the mapping like $φ: 2\mathbb{Z} → 3\mathbb{Z}$ and then we ...
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1answer
42 views

Homomorphisms and automorphisms on polynomial rings

I am trying to prove a series of propositions: Given any homomorphism p from $\mathbb{R}$[X] to $\mathbb{R}$[X], show that it is equal to $\phi_g$ for a unique g in $\mathbb{R}$[X], with $\phi_g$(f) =...
2
votes
0answers
27 views

Classifying homomorphisms on polynomial rings with real coefficients.

Show that every homomorphism $\mathbb{R}$[X] $\rightarrow$ $\mathbb{R}$[X] can is equal to $φ_g$ for a unique g $\in$ $\mathbb{R}$[X], given by $φ_g(f)$ = $f(g(X))$ My guess for any homomorphism $h$, ...
0
votes
2answers
47 views

Proving a ring-homomorphism using a group-homomorphism

Let f : R → R' be a group homomorphism. Show that the induced map φ : R[x] → R'[x], where φ(anxn + . . . + a0) = f(an)xn + . . . + f(a0), is a ring homomorphism. I know that &#...
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vote
2answers
42 views

How to prove that $Ф(1) = 1'$ if $R'$ is an integral domain?

Since $R'$ is an integral domain , $Ф(b) = Ф(b.1) = Ф(b).Ф(1) = Ф(1).Ф(b)$ .But i can prove this only for those $r'∈R'$ for which there $∃r∈R$ such that $Ф(r)=r'$.
3
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1answer
30 views

Field Extensions and Injectivity

Here is a theorem from Judson and part of its proof: I am concerned about the part that is squared in red. Why do we have to show that the function is injective? Is this a common tactic in finding an ...
4
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2answers
57 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 ...
2
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3answers
59 views

Ring Homomorphism: $\mathbb{Z}[x] / (f(x)) \to \mathbb{Q}$

Let $f(x) \in \mathbb{Z}[x]$. Prove that $f(x)$ has a root in $\mathbb{Q}$ iff there is a ring homomorphism from $\mathbb{Z}[x]/(f(x)) \rightarrow \mathbb{Q}$. I tried using a homomorphism from $\...
3
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2answers
59 views

The number of surjective ring homomorphism from $\mathbb{Z}[i]$ to $\mathbb{F}_{11^2}$.

Find the number of surjective ring homomorphism from $\mathbb{Z}[i]$ onto $\mathbb{F}_{11^2}$. If such a surjective ring homomorphism exists with kernel $(a+bi)$, then $\mathbb{Z}[i]/(a+bi)\cong\...
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2answers
24 views

Proving that an inverse ring homomorphism of an ideal is an ideal?

Say I have some function $f:R \rightarrow S$ such that $f$ is a ring homomorphism and $J$ is an ideal of $f$. $I = f^{-1}(J)$ is an ideal of $R$, but I don't really understand why. $J$ being an ideal ...
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1answer
82 views

What is an inclusion in a ring homomorphism?

I was reading some notes on Ring theory that I found online and it says this : If $R$ is any ring and $S ⊂ R$ is a subring, then the inclusion $i: S → R$ is a ring homomorphism. I don't know ...
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1answer
48 views

For a ring homomorphism prove that $f(a^{-1})=[f(a)]^{-1}$ [closed]

$R$ and $S$ are commutative rings and f is a ring homomorphism $f:R\rightarrow S.\ $ For $a\in R$ prove that $$f(a^{-1})=[f(a)]^{-1}$$ Please tell me how can I start this proof.
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1answer
63 views

Schur's Lemma and $Z(G)$

Let $Z(G)$ be the centre of G. Let $V$ be an irreducible $G-$space with matrix representation $\rho_v$. Let $z \in Z(G)$, then I'm trying to show that $\rho_v(z)$ is multiplication by a root of ...
3
votes
2answers
38 views

Prove or disprove: The image of a ring homomorphism $\phi:R\to S$ is an ideal in $S$.

Prove or disprove: The image of a ring homomorphism $\phi:R\to S$ is an ideal in $S$. I only see examples where they use the image of an ideal, but I don't think this is the case for my question. ...
1
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2answers
29 views

Showing this map is injective

Define a map $$\phi : \mathbb{Q}[x] \rightarrow \mathbb{Q}[x]$$ by $\phi(g(x)) = g(x-2)$ It is straightforward to show that this is a homomorphism, but I'm having trouble showing it's surjective &...
1
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1answer
39 views

Proving this map is injective

Let $R$ be a ring and let $J$ be a left ideal of $R$. Let $s \in R$. We can make the module $R/J$. Let $\DeclareMathOperator{\ann}{ann}\ann_R(t + J) = {\{r \in R : rt + J = 0}\}$ be the annihilator. ...
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1answer
136 views

Ring Homomorphism from $\mathbb{Z}$ to a Field $F$ [duplicate]

I have so far shown that $\phi: \mathbb{Z} \to F$ is a unique ring homomorphism if $\phi(n)=n$ and that $ker(\phi) = (n) = \{nm|m \in \mathbb{Z}\}$ with $n = 0$ or $n=p$ with $p$ a prime number. ($n$ ...
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0answers
8 views

Homomorphism between semirings with opposite precedence of the binary operations

Suppose, $(R, \cup, \cap)$ and $(S, \cap, \cup, )$ are two different semirings with $\cup$ as additive operation and $\cap$ as multiplicative operation in $R$, while the same operations $\cap$ and $\...
2
votes
1answer
40 views

An example of a ring homomorphism

Before stating my question, let me recall some preliminaries in rings (especially noncommutative). Recall that for a noncommutative ring $R$, $‎\textbf{B}‎(R)=\{e\in Z(R): e^2=e\}$. $\textbf{...
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votes
2answers
42 views

Is there a function with these properties?

I am looking for a function $\phi$ with the properties described below: $\phi$ : $\mathbb{Z}_q^{n \times m} \rightarrow \mathbb{Z}_q^{n \times n}$. $\phi(A) - \phi(B) \neq 0$ and is invertible if $A \...
0
votes
1answer
40 views

Existence of an epimorphism of the localization to a quotient ring

I'm reading a book about Commutative Algebra from my library and I'm stuck in a problem about localization. Here I put the statement and my attemps. Let $M$ be a maximal ideal of a domain $R$ and let ...
2
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0answers
31 views

There is equivalence of norms and equivalence of metrics. Is there equivalence of order relations?

Two norms $||·||_1$ and $||·||_2$ over a space $X$ are equivalent iff there exist positive $c, C\in \mathbb{R}$ such that for all $x\in X$ $c||x||_1\leq ||x||_2\leq C||x||_1$. Two metrics $d_1$ and $...
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0answers
36 views

Inverse image of maximal ideal in quotient ring

I need to prove this: "Let $R$ be a ring, $I \subseteq R$ an ideal and $\phi : R \to R/I$ the quotient homomorphism. If $J \subseteq R/I$ is a maximal ideal of $R/I$, then $\phi^{-1}(J) $ is a maximal ...
0
votes
0answers
53 views

Determine the number of ring homomorphisms $\mathbb{Z}_{(13)}\rightarrow \mathbb{Z}_{13}$.

$\mathbb{Z}_{(13)}$ is the set of all reduced fractions where the denominator is a power of $13$. I have absolutely no idea how to do this so any help would be appreciated.
2
votes
2answers
148 views

Short proof of $\mathbb{Q}[x,y]/\langle x^2+1, y^4-2\rangle \equiv\mathbb{Q}[\sqrt[4]{2}, i]$

I am looking for an indirect proof of $$E = \mathbb{Q}[x,y]/_{\langle x^2+1, y^4-2\rangle}\cong\mathbb{Q}[\sqrt[4]{2}, i],$$ much preferably using module homomorphism theorems. To be more specific, ...
3
votes
1answer
78 views

Homomorphism from $\mathbb R^2\to \mathbb C$

Is it possible to define a surjective ring homomorphism from $\mathbb R^2$ onto $\mathbb C$? The multiplication defined on $\mathbb R^2$ is as follows: $(a,b)(c,d)=(ac,bd)$
4
votes
1answer
77 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.
1
vote
2answers
49 views

Regarding Banach algebra homomorphism

Let $A$ be a Banach algebra with identity $e_A$ and let $B$ be a Banach algebra with identity $e_B$. Let $\phi:A\longrightarrow B$ be a Banach algebra homomorphism i.e $\phi$ is linear and $\phi(ab)=\...
6
votes
4answers
123 views

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 ...
2
votes
1answer
56 views

Which max ideal/kernel of this homomorphism does this correspond to

Let $K$ be a field. Let $K[X]$ be a (nice) algebra over a field corresponding to an affine variety. For simplicity say it is $K[x_1,\dots,x_m]$ (usual polynomial ring). Then "points" of $X$ are ...