For questions about ring homomorphisms, a function between two rings which respects the structure.
Questions tagged [ring-homomorphism]
587 questions
3
votes
0 answers
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 an integral domain and $A \neq B$ then every…
user1142333
- 91
2
votes
1 answer
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 $\mathbb{R}$ to
$ ϕ(x^2 + 1) = ϕ(0)$
$ ϕ(x^2)+ ϕ(1) =…
user12938120
- 81
- 7
1
vote
1 answer
Show that $\{x \in R | f(x) = a'\} = a + Ker(f)$ as following.
Let $f:R \to R'$ be a homomorphism and $a\in R, a' \in R'$ such that $f(a)=a'$. Show that $\{x \in R | f(x) = a'\} = a + Ker(f)$.
I have tried find Ker(f) and stuck at $Ker(f) = \{x\in R | f(x) = 0_{R'}\} = \{x \in R | a' = 0_{R'}\}$.
What's…
lap lapan
- 2,188
1
vote
0 answers
Identifying whether or not that the map from $\mathbb{Z}/8\mathbb{Z}$ to $\mathbb{Z}/4\mathbb{Z}$ is isomorphism.
Let $f$ be a map from quotient ring from $\mathbb{Z}/8\mathbb{Z}$ to
$\mathbb{Z}/4\mathbb{Z}$ defined by
$f(z+8\mathbb{Z}) = z+4\mathbb{Z}$.
Check that whether or not that $f$ is isomorphism.
My attempt: (Edited)
For homomorphism,
let…
lap lapan
- 2,188
1
vote
1 answer
Are $10 \mathbb Z$ and $20 \mathbb Z$ isomorphic to each other or not as rings?
Are $10 \mathbb Z$ and $20 \mathbb Z$ isomorphic to each other or not as rings?
Since both of $10 \mathbb Z$ and $20 \mathbb Z$ are cyclic groups. So if some isomorphism $f$ between them do exist then either $f(10)=20$ or $f(10)=-20$. First let us…
user251057
1
vote
0 answers
Construct a bijection explicitly.
Construct a bijection $Hom_{\mathbb C}(\mathbb C[x,y]/(xy-1),\mathbb C) \to \mathbb C - \{0\}$.
Here the similar question lies but I can not find the exact bijection which is explicitly defined.
I considered the mapping $\phi_{a,b} \mapsto {a \over…
Mini_me
- 2,165
1
vote
1 answer
The number of non-trivial ring homomorohism from $Z_{20}$ to$ Z_{44}$
Consider $Z_{20}$ and $Z_{44}$ as ring modulo $20$ and $44$ respectively.Then number of non-trivial ring homomorohism from $Z_{20}$ to $Z_{44}$ is?
Arib khan
- 129
0
votes
1 answer
multiplication or composition?
I am studying the ring homomorphism of the following function:
If $f: M \to N$ is a smooth function, then $$f^*: C^{\infty}(N) \to C^{\infty}(M), \textbf{ defined by } \phi \mapsto \phi \circ f$$ is a ring homomorphism.
I know that the given…
Brain
- 1,003
0
votes
2 answers
Ring Homomorphism Question
Let $f:F_1 → F_2$ be a ring homomorphism between fields $F_1, F_2$.
(a) Show that if $f(1)=0$ then $f=0$.
(b) Show that if $f(1)\ne0$ then $f$ is injective.
Hi, I'm not too sure how to do this question. Any help would be greatly appreciated :)
Karina Lam
- 11
0
votes
0 answers
Would there be homomorphism?
$H=\left\{ \begin{bmatrix} a & b \\ 0 & c \end{bmatrix}:a,b,c\in Z\right\}$ get the ring $\varphi :H\rightarrow Z$ , $\varphi \left( \begin{bmatrix} a & b \\ 0 & c \end{bmatrix}\right) =c$ Does the transformation become a ring homomorphism?…
Yusuf Kanat
- 69
0
votes
2 answers
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 if it is homomorphism, surjective and instead of…
Haus
- 61
0
votes
1 answer
Problem in solving a question of field homomorphism.
The question is :
Let $F$ and $F'$ be two finite fields with nine and four elements respectively.How many field homomorphisms are there from $F$ to $F'$?
My effort:
Let us consider a homomorphism $f : F \to F'$. Now since ker $f$ is an ideal…
user251057
-1
votes
1 answer
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 say $φ(2) = 3k, k∈Z$.
How to start in this case…
Haus
- 61