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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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Ring Homomorphism Identities [on hold]

Suppose $f: R\rightarrow S$ is a ring homomorphism such that $R$ is a ring with identity $1_R$. Prove that $f(1_R)$ is the multiplicative identity in $f(R)$ I am stumped on this one. Learning ...
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Finding All Ring Homomorphisms from $\Bbb Z_m$ to $\Bbb Z_n$

I was working through Abstract Algebra : A Geometric Approach (by Theodore Shifrin) and I came across this exercise that I am struggling with. ($ℤ_m$ means integers mod m) Find All Ring ...
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Image of a subgroup is subgroup under homomorphism.

Image of a subgroup under group homomorphism is a subgroup of "codomain." Image of a normal subgroup under group homomorphism is a normal subgroup of "range." Image of a subring under ring ...
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Find all ring homomorphisms [duplicate]

Find all the ring homomorphisms $f$ : $\mathbb{Z}_6\to\mathbb{Z}_3$. definition of ring homomorphism: The function f: R → S is a ring homomorphism if: 1) $f(1)$ = $1$ 2) $f(a+b)$ = $f(a)$ + $f(b)$ ...
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Why is $k[S_{r+1},\ldots,S_n]\to k[X_1,\ldots,X_n]/I$ injective?

I'm reading Qing Liu's Algebraic Geometry and Arithmetic Curves. The proof of the following proposition says that (a) and (b) implies (c). But I can't prove (c) from (a) and (b). Proposition 1.11. ...
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homomorphism question

Check $p(x) + p(-x) ∈ P^e$ for every $p(x) ∈ \mathbb{R}[x]$. Check that the map $Ψ:\mathbb{R}[x]\to P^e$ given by $Ψ(p(x))=(p(x)+p(-x))/2$ is a linear map. Further check that $Ψ^2=Ψ$. Determine ...
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How to find initial object of the category of pointed rings?

I have the category of pointed rings. Objects are all pairs $(R, r)$ where $R$ - ring (with 1) and r is the element of R. Morphisms are homomorphism of rings. Morphism $(R, r) \longrightarrow (R', r')$...
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Proof $\varphi:\mathbb{Z} \rightarrow \mathbb{Z}/p_1 ^{a_1}\mathbb{Z} \times…\times\mathbb{Z}/p_3 ^{a_3} \mathbb{Z}$ is surjective.

I want to prove a homomorphism $\varphi:\mathbb{Z} \rightarrow \mathbb{Z}/p_1 ^{a_1}\mathbb{Z}\times \mathbb{Z}/p_2 ^{a_2}\mathbb{Z}\times\mathbb{Z}/p_3 ^{a_3} \mathbb{Z}$ is surjective, where $p_1,...
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Prove that the only ring homomorphism from $\mathbb Z$ to $\mathbb Z$ is Identity mapping

Suppose there exists some other ring homomorphism than the identity mapping, $f$ from $\mathbb Z$ to $\mathbb Z$. First I observe $f$ sends 0 to 0, and 1 to 1, and permutes some other elements in $\...
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#Hom$(\mathbb{Z}[i], \mathbb{Z}/85\mathbb{Z})$

I'm trying to determine the number of ring homomorphisms from $\mathbb{Z}[i]$ to $\mathbb{Z}/85\mathbb{Z}$. My reasoning is as follows: We know that for all $f \in \text{Hom}(\mathbb{Z}[i], \mathbb{Z}...
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Why $\mathbb{Z}[\sqrt{-5}]\cong\mathbb{Z}[X]/(X^2+5)$ and $\mathbb{C}\cong \mathbb{R}[X]/(X^2+1)$?

This is a very elementary question. In Matsumura's book "Commutative ring theory", I've found the following isomorphism: $$\mathbb{Z}[\sqrt{-5}]\cong\mathbb{Z}[X]/(X^2+5).$$ As a quotient ring is ...
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A homomorphism on a nontrivial commutative ring with trivial unit group

Suppose $R \neq \{0\}$ is commutative and satisfies $R^\times = \{1\}$. I've shown that this implies that $\operatorname{char}R=2$ (by showing that $-1 = 1$). Now consider the homomorphism $f \in \...
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When does $\mathbb{Z}[\zeta_m]$ contain divisors of $2$ (besides units)?

Or equivalently, in which $\mathbb{Z}[\zeta_m]$ is $2$ reducible? And how does one construct any such divisors? $\bullet\ \textbf{My attempt}$ The smallest example seems to be $m=4$ with $2=(1+i)(1-...
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$x\mapsto x^p$ with $\operatorname{char}(F)=p>0$ is a ring homomorphism [closed]

Let a field $F$ with $\operatorname{char}(F)=p>0$. Let a map $f:F\to F$ defined be $x\mapsto x^p$. Show that $f$ is a ring homomorphism. Obviousely $0\mapsto 0$ and $(xy)^p=x^py^p$. How can I show ...
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Ring homomorphism from $\mathbb{Z}$ to integral ring.

From Lang, Undergraduate Algebra: Theorem 3.2 Suppose that $R$ is an integral ring (commutative ring without zero-divisors and such that $1\ne 0$). Then the integer n such that $\mathbb{Z}/n\mathbb{Z}...
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Finding an isomorphic subring of matrices

I'm struggling a fair amount with this exercise: Find a subring of $M(2,\mathbb{Q})$ which is isomorphic to a) $\mathbb{Q}$ x $ \mathbb{Q}$ b) $\mathbb{Q}$ c) $\mathbb{Q}[x]$/$x^2$ Now I know a ...
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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) ...
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field extension being algebraic is equivalent to every $K-$ algebra being an automorphism.

For a field extension $L\vert K,$show that the following statements are equivalent : $(i)$$L\vert K$ is algebraic. $(ii)$ For every $E\in${$E:E$ is a field with $K\subset E\subset L$},...
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Showing a map from $\Bbb Q$ to the field of fractions of $\Bbb Z 1_{R}$ is well defined.

Suppose that I have a field $R$ with characteristic $0$. Let $Q= \{uv^{-1} | u,v \in \Bbb Z 1_{R}, v \neq 0 \}$. I want to show that the map $f :\Bbb Q \rightarrow Q$ defined as $f(n/m) = (n1_{R})(m1_{...
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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 $a \notin F \cup I$....
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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 ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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“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 ...
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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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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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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 $f,g,h\in R$: $(f+g)\circ h=f\circ h +g\circ h$ $...
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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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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 $...
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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 $\mathbb{Z_7}$ ...
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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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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 $a \in R$ ...
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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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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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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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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) =...
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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 $h$, ...
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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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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 $Ф(r)=r'$.
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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 ...
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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 ...
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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 $\...
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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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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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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 ...