-4
votes
1answer
38 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
34 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
62 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
183 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
158 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
92 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
777 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
148 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
91 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 ...
1
vote
3answers
182 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
331 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
436 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 ...