Questions tagged [ring-homomorphism]

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

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30 views

Proof Verification: Hom(Z[x],S)=S (as rings)

How to prove Hom(Z[x],S)=S (as rings), where S is any ring? My attempt: took an element b in S, defined a map , b: Z[x]-> S which maps f(x) to f(b). Clearly b is a ring homomorphism, hence we ...
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1answer
16 views

Realizing a module structure with endomorphisms

I am required to show that there is a match between seeing an abelian group $M$ as an $R-$Module and homomorphisms $R\to End(M)$. Where the multiplication in $End(M)$ is defined as : $f\cdot g=g\circ ...
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2answers
22 views

Examples of homomorphisms

Could someone give examples homomorphisms of rings f: R->S and g: S->T such that gof is a monomorphism but g is not? I tried with the maps from Z (a->na) but can't think of a map such that ...
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1answer
24 views

List all additive morphisms $f: Z_m \to Z_n$, with $Z_n$ the integers modulo $n$.

I'm new to abstract algebra and I'm trying to derive the general form of any function $f: Z_m \to Z_n$, such that $f$ is an additive (homo-)morphism, and where $n, m \in N$, $N$ being the set of ...
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1answer
29 views

Uniqueness of homomorphism -universal property

If $\varphi: R\to A$ is a homomorphism such that for a subset $S$, $\varphi(s)$ is inevertable for every $s\in S$ then there is a unique homomorphism $\varphi^\prime:S^{-1}R\to A$ such that $\varphi = ...
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1answer
18 views

Isomorphic $K[x]$-modules have equal characteristic polynomials

Let $K$ be a field, $V,V'$ be finite dimensional $K$-vector spaces and $A\in \text{End}_K(V),A'\in \text{End}_K(V').$ Regard $V$ (respectively $V'$) as $K[x]$-modules with respect to $A$(resp. $A'$.) ...
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1answer
15 views

Finite-Type Algebra Property Under Change of Base

I was wondering how to prove the following assertion. Any help would be appreciated! Suppose we have to ring homomorphisms: $f: A \rightarrow B$ and $g: B \rightarrow C$. Then $C$ being finite type ...
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39 views

Non-trivial homomorphism from $\mathbb{Z}[x]/\langle x^n-1 \rangle$ to $\mathbb{Z}_m[x]/\langle x^n+1 \rangle$?

Is there any non-trivial homomorphism from $R_1=\mathbb{Z}[x]/\langle x^n-1 \rangle$ to $R_2=\mathbb{Z}_m[x]/\langle x^n+1 \rangle$, for all $m>1$ (no necessarily prime)? (Note that the elements in ...
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1answer
42 views

Primary decomposition of an ideal and its extension

I'm trying to solve a problem in Sharp's Steps in Commutative Algebra, to be precise Exercise 4.22 which states the following: Let $f:R \rightarrow S $ be a surjective homomorphism of commutative ...
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1answer
28 views

Subjectivity of evaluation homomorphism

I am trying to prove that the evaluation homomorphism: $ev_z:\mathbb{R}[X]\to \mathbb{C}, f\mapsto f(z)$ where $z=a+bi$ is surjective. To start with, I don't really understand how the map is ...
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1answer
47 views

$\varphi(u)$ is invertible in $R'$ iff no invertible element of $R$ belongs to $\ker \varphi$.

Assume the surjective ring homomorphism $\varphi: R \longrightarrow R'$, where $R$ is a ring with unity. Also let $u$ be an invertible element of $R$. Show that $\varphi(u)$ is invertible in $R'$ ...
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1answer
28 views

A module over PID [closed]

Let R be a PID and M be a module over R ⋀ cyclic. Then prove that ∀N⊂M; a submodule of M, N is cyclic. [My idea] ∃m∈M s.t. M=Rm (∵M is cyclic) p: R(= a module over R) → M, p(a) = am; module-hom ⋀ ...
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1answer
17 views

Find a ring homomorphism $\theta$ s.t. Ker $\theta = \mathbb{Z}_6 \times \{[0]\}$.

Find a ring homomorphism $\theta: \mathbb{Z}_6 \times \mathbb{Z}_{14} \to \mathbb{Z}_6 \times \mathbb{Z}_{14}$ for which Ker $\theta = \mathbb{Z}_6 \times \{[0]\}$. Attempt: I know that Ker $\theta =...
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2answers
41 views

For which ideal $I$ of $\Bbb Z[t]$ is $\mathbb{Z}[t]/I\cong\Bbb Z_{11}$? [closed]

Maybe for $I=(11,t-1)$ but i don't know how to prove it or if it is even right.
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1answer
33 views

How to prove that $(\Bbb{Z}[t]+t\Bbb{R}[t])/t\Bbb{R}[t]\cong\Bbb{Z}\cong\Bbb{Z}[t]/t\Bbb{Z}[t]\cong\Bbb{Z}[t]/(\Bbb{Z}[t]\cap t\Bbb{R}[t])$?

I already proved that $(\mathbb{Z}[t]+t\mathbb{R}[t])/t\mathbb{R}[t]\cong\mathbb{Z}[t]/(\mathbb{Z}[t]\cap t\mathbb{R}[t])$ with the first isomorphism theorem but i do not know how to continue.
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1answer
55 views

extending ring homomorphism into fields

Let $A$ be a subring of $B$ such that $B$ is integral over $A$. Show that every ring homomorphism $f:A\rightarrow K$ with $K$ an algebraically closed field can be extended to a ring ...
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2answers
35 views

$(\mathbb{Z}/18\mathbb{Z})/(6\mathbb{Z}/18\mathbb{Z})\cong\mathbb{Z}/6\mathbb{Z}$ Proof

Use the homomorphism theorem or the first or/and second isomorphism theorems to show that $(\mathbb{Z}/18\mathbb{Z})/(6\mathbb{Z}/18\mathbb{Z})\cong\mathbb{Z}/6\mathbb{Z}$. I was wondering if it is ...
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1answer
73 views

Characterization of Injective rings homomorphism from $\mathbb{Z}_m$ to $\mathbb{Z}_n$

Show that there is an injective ring homomorphism $f:\mathbb{Z}_m \rightarrow \mathbb{Z}_n$ if and only if $m\mid n$ and $\frac{n}{m}$ is relatively prime with $m$. In one direction, was not ...
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0answers
19 views

Is there any way to visualize Ring structure using Cayley digraph?

In Nathan Carter's Visual group theory there is a nice description of how to explore group structure through Cayley digraph and how the structures reflect themselves in Cayley tables.For example ...
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0answers
26 views

Confusion about faithful homomorphism

Let $Q=\oplus_{u \in V} Ru$ be a free left $R$-module on a set $V$. Let also consider $End(Q)$, my first confusion is about how we can see $End(Q)$ as a ring, and then how can we check that such this ...
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2answers
57 views

Is the ring $3\mathbb Z$ a ring homomorphic image of the ring $2\mathbb Z$.

Is the ring $2\mathbb Z$ isomorphic to the ring $3\mathbb Z$ ? Solution: Let if possible,$\phi:\mathbb {2Z\to 3Z}$ be a ring isomorphism. Then $\phi$ is a group isomorphism between the additive ...
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0answers
24 views

Would there be homomorphism?

$H=\left\{ \begin{bmatrix} a & b \\ 0 & c \end{bmatrix}:a,b,c\in Z\right\}$ get the ring $\varphi :H\rightarrow Z$ , $\varphi \left( \begin{bmatrix} a & b \\ 0 & c \end{bmatrix}\right) ...
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0answers
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Help determine what kind of mapping

If I have that for any equation of the form $$(2^n-1)(1+x+x^2)(1+y+y^2)(1+z+z^2)-2^n(x^2y^2z^2)$$ and given any solution in positive integers $x_0,y_0,z_0$ a map $f(x_0,y_0,z_0) \longrightarrow (2^{n},...
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0answers
13 views

Homomorphis of the algebra of measurable fucntions

Let $(X,\Omega,\mu)$ be a measure space and $L_0(X)$ be the algebra of measurable functions. Suppose $T\colon L_0(X)\to L_0(X)$ injective homomorphism that is linear and $T(fg)=T(f)T(g)$ for all $f,g\...
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0answers
21 views

Show $f$: $Z_{mn}\mapsto Z_{m}$ $\times$ $Z_{n}$ by the rule $f(x\pmod{mn}) = (x\pmod{m}, x\pmod{n})$ is a homomorphism.

I defined a function from two rings $f$: $Z_{mn} \mapsto Z_{m}$ $\times$ $Z_{n}$ as $f(x\pmod{mn}) = (x\pmod{m}, x\pmod{n})$ and I want to show that it is a homomorphism. Observe that for any $x,y \...
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1answer
28 views

Prove there cannot be a ring homomorphism $ ϕ : \mathbb{C} → \mathbb{R}$

Is my proof to the question correct? In $\mathbb{C}$ we have that $i$ is the solution to $x^2 + 1 = 0 $. Thus if a homomorphism exists from $\mathbb{C} \to \mathbb{R}$ there is a solution in $\mathbb{...
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28 views

Prove $Ra/Rab$ $\cong_R$ $R/Rb$ , with $R$ a domain and $a,b$ not zero

I posted a question yesterday on how to prove the following :" $Ra/Rab$ $\cong_R$ $R/Rb$ " and the answer I got said the homomorphism $\phi :$$R$ $\rightarrow$ $Ra/Rab$ : $x$ $\mapsto$ $ax + Rab$ ...
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2answers
44 views

Field homomorphism is onto [closed]

Let $F$ be a field and $A:F \to F$ be a homomorphism. Show that $A$ is onto. Abstract algebra proof. This question very important to me.
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1answer
22 views

Are order-preserving field embeddings unique?

Let $K$ be an ordered field with an embedding into $\mathbb R$, $$f:K\hookrightarrow\mathbb R,$$ where $f$ is order preserving. Is $f$ unique? Follow up from this question of mine (same question ...
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1answer
27 views

Are field embeddings unique?

Sorry if this is a simple question, as I'm not well versed in field theory. Suppose a field $K$ has an embedding into $\mathbb R$: $f:K\hookrightarrow\mathbb R$. Is $f$ unique? And if $\mathbb R$ is ...
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1answer
28 views

Question about rings homomorphism

I need to prove the following result Let $p$ and $q$ be 2 different primes. There isn´t any homomorphism from $\mathbb{Z}_p$ to $\mathbb{Z}_q$ or from $\mathbb{Z}_q$ to $\mathbb{Z}_p$ What I've ...
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2answers
30 views

Definition of homomorphism

If $\varphi: K \to K'$ is a map of Integral Domains st. $\varphi(xy)=\varphi(x) \varphi(y)$ and $\varphi(x+y)=\varphi(x) + \varphi(y)$. Then $\varphi(0)=0'$ as $\varphi(0)=\varphi(0+0)=\varphi(0)+\...
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2answers
58 views

Cardinal number of $U(\Bbb Z / 4 \Bbb Z \times \Bbb Z/5 \Bbb Z)$ and zero divisors of $\Bbb Z / 45 \Bbb Z \times \Bbb Z / 27 \Bbb Z$

I'd like to know how many units does $ \Bbb Z / 4 \Bbb Z \times \Bbb Z / 5 \Bbb Z $ has. And how many zero divisor has $ \Bbb Z / 45 \Bbb Z \times \Bbb Z / 27 \Bbb Z $. I think I should work with $\...
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1answer
58 views

Some Properties about the Characteristic of a Ring

Define characteristic of a ring $R$ as the natural number n such that $n\mathbb{Z}$ is the kernel of the unique ring homomorphism from $\mathbb{Z}$ to $\mathbb{R}$, which is given by $$ \begin{array}...
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2answers
40 views

Characteristic of a ring: $\ker(\varphi) = (n)$.

I've been asked to prove that there is exactly one ring homomorphism from the ring of integers to any ring with unity. And I proved that let $R$ be a ring with unity and $$ \begin{array}{rccl} \...
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2answers
34 views

Existence of ring homomorphism from $\mathbb{Z}[[x]] \to \mathbb{Z} $ [closed]

Let $\mathbb{Z}[[x]]$ denote the ring of formal power series with coefficients in $\mathbb{Z}$. Can there exist a ring homomorphism $$\Phi : \mathbb{Z}[[x]] \to \mathbb{Z}$$ such that $\Phi $ sends $...
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1answer
36 views

Is this proof on ideal maximality correct?

The exercise is: Given that $ \phi : R \to S $ is an onto ring homomorphism, let $ B $ be a maximal ideal of $ S $. Prove that $ A = \phi^{-1} (B) $ is a maximal ideal of $R$. Ok, I previously ...
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1answer
31 views

If $\varphi:R\to S$ be a surjective ring homomorphism then $\varphi(R^*) $is not always $S^*$ [closed]

I don't know any example. Whatever examples I know the statement holds true
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2answers
44 views

Direct Product of a Clean Rings

Commutative rings whose elements are a sum of an unit and idempotent by Anderson & Camillo (2002) Definition 1. A commutative ring $R$ is a clean ring if every element $x\in R$ can be written in ...
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1answer
63 views

Existence of Ring homomorphism from Formal Power series ring to a ring.

Let $R$ be a ring, and $S$ be a subring of $R$. Denote $S[[x]]$ for a ring of formal power series with coefficients in $S$. Let $\alpha \in R$ be a unit, such that $\alpha \notin S$. Can there exist a ...
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1answer
26 views

Ideal of quotient ring properly contains the ideal used to form the quotient ring

The proofs I have seen of "N is a maximal ideal in a ring R iff R/N is a simple ring" involve a step using the canonical homomorphism $\phi$ and an ideal of $R/N$. For example, suppose $\phi: R \...
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1answer
39 views

Homomorphism from a ring with unity to a integral domain maps unity to unity?

I know a similar question was asked here. But my exercise is asking me to do without a hypothesis. If R and R' are rings with unity(denote $1$ and $1'$ for the $R$ and $R'$ identities, respectively)...
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1answer
53 views

Ring homomorphism between a ring R with an identity and an integral domain S is either 0 or maps $1_R$ to $1_S$ [closed]

I have been reading my Rings & Module course notes and the lecturer has written a proposition with certain properties of ring homomorphisms such as $\phi(0_R) = 0_S$ and some other without any ...
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1answer
44 views

Homomorphism of Polynomial Domains induced by a homomorphism of the ring of coefficients (G1)

This question originates from Pinter's Abstract Algebra, Chapter 24, G1. Let $A$ and $B$ be rings and let $h: A\rightarrow B$ be a homomorphism with kernel $K$. Define $\bar{h}: A[x]\rightarrow ...
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1answer
18 views

Homomorphism of Polynomial Domains induced by a homomorphism of the ring of coefficients (G7)

This question originates from Pinter's Abstract Algebra, Chapter 24, G7. Let $h:\mathbb{Z}\rightarrow\mathbb{Z}_n$ be a homomorphism with kernel $K$. Define $\bar{h}: \mathbb{Z}[x]\rightarrow\...
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0answers
6 views

Can we define a mapping from the set of graphs to the set of adjacency matrices?

Graph union: the union of two graphs $G_1=(V_1, E_1)$ and $G_2=(V_2, E_2)$ is defined and denoted by $G_1\cup G_2=(V_1\cup V_2, E_1\cup E_2)$. Graph intersection: the intersection of two graphs $G_1=(...
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0answers
32 views

Mapping rational numbers to a fixed real number by a homomorphism

Let $x$ be a fixed real number and let $\mathbb{Q}$ be the field of rational numbers. It is known that $\mathbb{R}$ is an extension field of $\mathbb{Q}$ and thus there exists some homomorphisms. Can ...
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1answer
19 views

Image of a semiring homomorphism, which is an ideal.

Let $\phi: (R, +, \cdot)\rightarrow (S, \oplus, \star)$ be a semiring map. Then $\operatorname{im}\phi$ is a sub semiring of $S$. In general, it is not an ideal. I look for an example in which $\...
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0answers
64 views

Prime ideals of ring $R = \{(a, b) \in \mathbb{Z} \times \mathbb{Z} | \, a \equiv b\, \mod 5 \}$.

This question have two parts. I was able to do the first part. Consider the ring $R = \{(a, b) \in \mathbb{Z} \times \mathbb{Z} | \, a \equiv b\, \mod 5 \}$ a) Show that the homomorphism ...

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