Questions tagged [ring-homomorphism]
For questions about ring homomorphisms, a function between two rings which respects the structure.
594
questions
3
votes
2
answers
113
views
Homomorphism between $\mathbb{Z}[i]$ and $\mathbb{ Z/(p)}$
I'm trying to construct a non-zero ring homomorphism between $\mathbb{Z}[i]$ and $\mathbb{ Z/(p)}$. The question is for what $p$ is it possible. I managed with one direction:
$f(1) = f(-i^2)=-(f(i))^2 ...
- 95
0
votes
1
answer
50
views
Let $\varphi: K_1 \rightarrow K_2$ be a ring homomorphism and $M$ a left $K_2$-module.
Let $\varphi: K_1 \rightarrow K_2$ be a ring homomorphism and $M$ a left $K_2$-module.
(a) Prove that $M$ becomes a left $K_1$-module if we introduce the operation of multiplication by elements from $...
- 45
-4
votes
0
answers
15
views
Is the homomorphic image of a ring an ideal of the co domain? [closed]
Q. If f be a homomorphism from a ring R into a ring R'. Then show that f(R) is an ideal of R'.
As per my knowledge, it is not possible. I want a very clear idea about this question and the solution.
...
- 1
0
votes
1
answer
57
views
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 ...
- 3,065
0
votes
2
answers
41
views
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 $\mathbb{...
- 15
0
votes
1
answer
37
views
LAnn$_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 ...
- 917
2
votes
1
answer
63
views
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 $\phi$...
- 31
1
vote
0
answers
58
views
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\...
- 1,277
2
votes
1
answer
57
views
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. ...
- 329
0
votes
1
answer
63
views
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 ...
- 1,131
2
votes
1
answer
84
views
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{\...
- 795
1
vote
1
answer
58
views
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,\...
- 469
-3
votes
1
answer
82
views
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 ...
- 3,127
1
vote
2
answers
116
views
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 ...
- 37
1
vote
1
answer
63
views
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 ...
- 111