3
votes
2answers
28 views

group of module homomorphisms is a module

I am trying to solve the following problem: Let $M$ and $N$ be two left $A$-modules. Prove that $Hom_A(M,N)$ has a left $Z(A)$-module structure with: $(a.f)(m)=a.f(m)$. Show $Hom_A(A,N) \cong N$ as ...
0
votes
0answers
30 views

$\mathbb R[X,Y,Z]/\langle Y-X^5,Z-Y^5\rangle \cong \mathbb R[X]$

I am trying to show that the rings $\mathbb R[X,Y,Z]/\langle Y-X^5,Z-Y^5\rangle$ and $\mathbb R[X]$ are isomorphic. I am pretty lost on how to do this. I suppose that the idea is to exhibit an ...
0
votes
1answer
25 views

Ring morphisms, group ring problem

Prove that if $A$ is a ring and $G$ is a group, then the map $$Hom(\mathbb Z[G],A) \to Hom(G,\mathcal U(A))$$ which sends $f \rightarrow f|G$ is a bijection. First of all, I am having some problem ...
2
votes
1answer
39 views

Ring homomorphism $f: \mathbb C \to \mathbb C$ such that $f(\mathbb R) \subset \mathbb R$

Exercise Find all the ring homomorphisms $f: \mathbb C \to \mathbb C$ such that $f(\mathbb R) \subset \mathbb R$ My attempt at a solution In a previous problem I've already proved that the only ...
0
votes
1answer
20 views

Statements about ring homomorphisms and division rings

Problem Decide whether the following statements are false or not. 1) If $A$ is a commutative ring such that every ring homomorphism different from the null morphism $\phi:A \to A'$ is injective, ...
1
vote
3answers
82 views

Characterizing kernel of ring morphism

Let $K$ be a field and define a ring morphism $$\psi: K[x_1,x_2, \dots , x_n, y_1, y_2, \dots , y_n] \rightarrow K(x_1,x_2, \dots , x_n)$$ by $\psi(x_i) =x_i$ and $\psi(y_i) =\frac{1}{x_i}$. I ...
-4
votes
1answer
40 views

Question about nonisomorphic polynomial rings.

Let $n,k > 1$ be positive integers. Define the reduced polynomial rings : $g^k_n = \Bbb R[X_n]/(G^k_n(X_n))$ where $G^k_n$ is a real polynomial of degree $n$ (that keeps the ring reduced). (k is ...
2
votes
1answer
35 views

Isomophism between rings an two right ideals

Let I, J two right ideals of a ring R such that I+J =R. Show thath the direct sum of I and J is isomorphic to the direct sum of R and the intersection of I and J. Can anyone please give me at least ...
0
votes
2answers
67 views

How is this map a well-defined homomorphism?

If $f: R \rightarrow S$ is a homomorphism of rings with kernel $K$, and $I$ is an ideal in $R$ such that $I \subset K$. The hypothesis is that the map $\overline{f}: R/I \rightarrow S$ given by ...
3
votes
3answers
191 views

Isomorphisms and the Fundamental Homomorphism Theorem

Let $$ R=\left\{ \begin{bmatrix} a & b \\ 0 & a \end{bmatrix} : a,b∈ℝ\right\}⊂M_2(ℝ) $$ and $$ I=\left\{ \begin{bmatrix} 0 & b \\ 0 & 0 \end{bmatrix}: b∈ℝ\right\}. $$ Identify the ...
0
votes
1answer
87 views

Finding homomorphisms and kernels from a given ring R

Give the following rings $R$ and ideals $I$ find a ring $S$ and a homomorphism $f:R \rightarrow S$ with kernel $I$ i) $R=\mathbb{Q} [x], I=(x^{2}-2)R$ ii)$R=\mathbb{Z}[i], I=2R$ (Gaussian Integers) ...
1
vote
2answers
172 views

Is the ring $\mathbb{Z}_5[x]$ isomorphic to the ring of polynomial functions from $\mathbb{Z}_5$ to $\mathbb{Z}_5$?

Is the ring $\mathbb{Z}_5[x]$ isomorphic to the ring of polynomial functions from $\mathbb{Z}_5$ to $\mathbb{Z}_5$? If not, what is a good counterexample? If yes, how can we prove that there's a ...
3
votes
1answer
96 views

Does this morphism necessarily give rise to a finite extension of residue fields?

Let $f:X\rightarrow Y$ be a morphism of finite type of locally Notherian schemes. Let $x\in X$ and $y=f(x)$. Recall that $f$ is said to be unramified if the map of stalks $g:\mathcal O_{Y,y} ...
30
votes
4answers
793 views

$\mathbb C[X]/(X^2)$ is isomorphic to $\mathbb R[Y]/((Y^2+1)^2)$

This question led me to the following: Prove that $\mathbb C[X]/(X^2)$ is isomorphic to $\mathbb R[Y]/((Y^2+1)^2)$.
1
vote
3answers
153 views

Ring homomorphism question.

If $R$ is a ring, show that there is exactly one ring homomorphism $\phi: \mathbb{Z} \to R$. I can't grasp the idea that there can be only one ring homomorphism. Aren't there many (at least more than ...
-1
votes
1answer
95 views

Problem of monomorphism of rings [duplicate]

Let $A$ a ring, for each monomorphism $f:A^m \rightarrow A^n$, I don't know how to prove that $m\leq n$. I can't start the problem, I have no idea, help me please.
5
votes
5answers
1k views

In a ring homomorphism we always have $f(1)=1$? [duplicate]

Possible Duplicate: the image of $1$ by a homomorphism between unitary rings I'm studying the Atiyah's commutative algebra book and I realized that in the beginning of the book, the author ...
3
votes
5answers
622 views

Number of ring homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{28}$.

Question: Find the number of non trivial ring homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{28}$. ($f$ is not necessarily unitary, i.e., $f(1)$ need not be $1$.) Suppose $f$ is a ring ...
1
vote
3answers
214 views

Are monomorphisms of rings injective?

Let $R$ and $S$ be rings and $f:R\to S$ a monomorphism. Is $f$ injective?
3
votes
1answer
341 views

the image of $1$ by a homomorphism between unitary rings

let $R$ and $S$ be unitary rings and $\phi:R\rightarrow S$ a ring homomorphism. is the following correct: $\phi(1_R\cdot1_R)=\phi(1_R)\cdot\phi(1_R)$ so $\phi(1_R)(1_S-\phi(1_R))=0_S$ and so ...
6
votes
3answers
453 views

What are all the homomorphisms between the rings $\mathbb{Z}_{18}$ and $\mathbb{Z}_{15}$?

Any homomorphism $φ$ between the rings $\mathbb{Z}_{18}$ and $\mathbb{Z}_{15}$ is completely defined by $φ(1)$. So from $$0 = φ(0) = φ(18) = φ(18 \cdot 1) = 18 \cdot φ(1) = 15 \cdot φ(1) + 3 \cdot ...