# Questions tagged [ring-homomorphism]

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

513 questions
Filter by
Sorted by
Tagged with
45 views

### Such an Integral domain exists or not? [duplicate]

There exists an integral domain $R$ and a surjective homomorphism $R→R$ of rings that is not injective. True or False. This was asked in one of the Ph.D. Selection Exam. I was unable to understand how ...
20 views

### k field. What is the correct $n$ and the correct ideal $I\subset k[X_1,\dots ,X_n]$ such that $k[X^2]\simeq k[X_1,\dots ,X_n]/I$ as a $k$-Algebra?

First i thought about $n=1$ and $I=(x-x^2)$, but later i realized that this ideal is the kernel of the homomorphism $$\varphi:k[x]\rightarrow k\times k$$ $$f\mapsto(f(0),f(1))$$ So now i think the ...
• 162
1 vote
39 views

1 vote
31 views

### Does a contradiction occurs about a homomorphism on $C(\Omega)$?

In the page 601 of 'HOMOMORPHISMS OF BANACH ALGEBRAS'(Bade and Curtis) https://www.jstor.org/stable/pdf/2372972.pdf, we get an inequality \begin{align*} ||\nu(x_m - x_n)|| \geq \rho_{\mu(C(\Omega))}(\...
• 281
39 views

### Prove that if $\Bbb{Q} ≤ E$ and $\Bbb{Q} ≤ F$, then $φ(r) = r$ for all $r ∈\Bbb{Q}$

Suppose $E$ and $F$ are fields, and $φ : F → E$ is a ring homomorphism such that $φ(1) = 1$. I've shown that $φ$ is injective. But if I have $\Bbb{Q} \subset E$ and $\Bbb{Q} \subset F$, how do I show ...
• 85
45 views

### What is the kernel of this homomorphism?

Let $$q : \mathbb{C}[a_1,a_2,a_3,a_4] \rightarrow \mathbb{C}[u,v]$$ $$a_1\rightarrow u+3v,a_2\rightarrow 3uv+v^2, a_3\rightarrow v^2(3u+v), a_4\rightarrow uv^3$$ be a ring map. My goal is to ...
• 168
95 views

• 95
46 views

• 4,081
112 views

• 489
1 vote
30 views

### Homomorphisms between $\Bbb Z[S_3] \to \Bbb Z[\Bbb Z \backslash 2 \Bbb Z]$

Show that there are at most $4$ ring homomorphisms between $\Bbb Z[S_3] \to \Bbb Z[\Bbb Z \backslash 2 \Bbb Z]$. Here is what I did : We know that ring homomorphisms send inversible elements to ...
• 996
58 views

### Additive inverses in endomorphism ring

I'm working my way through a proof that, given a ring $\langle R, +, \cdot, \theta \rangle$, $\langle \text{End}(R), +, \circ, z \rangle$ is also a ring (where $z(x)=\theta$ for all $x \in R$). ...
• 183
38 views

### 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 ...
• 1,161
52 views

• 301
45 views

### Show that there is one and only one homomorphism such that the tensor product diagram commutes

Proposition Let $M_1,...,M_n$ de modules over a commutative ring $R$. If $(P,p)$ is a tensor product of $M_1,...,M_n$, then, for every multilinear function $q:M_1\times\cdot\cdot\cdot\times M_n\to Q$, ...
• 301
1 vote
70 views

• 597
1 vote
32 views

### Prove: $F[t]/H[t]$ is isomorphic to $(F/H)[t]$

Let $F$ a commutative and unitary ring, and also $F$ is an integral domain. Let $H$ be a proper ideal of $F$. I want to prove: $F[t]/H[t]$ is isomorphic to $(F/H)[t]$ So my idea is to use the ...
• 597
38 views

86 views

### Under what conditions will the ring homomorphism $\phi : R \to S$ satisfy the following results about prime and maximal ideals?

Let $R$ and $S$ be rings and $\phi : R \to S$ be a ring homomorphism. Here, I am considering that $R$ and $S$ don't necessarily have multiplicative identities. MOTIVATION : I know that the pre-image ...
26 views

### Isomorphism of enveloping algebra homomorphisms to $k$-homomorphisms?

Given commutative ring $k$, associative $k$-algebra $A$, and $A$-bimodule $M$, we can define the enveloping algebra $A^e:=A\otimes A^{op}$, where $A^{op}$ is the algebra $A$ but with multiplication ...
24 views

### Why is this homomorphism surjective?

I have the following homomorphism between quotint rings: $$\mathbb{Z}[\sqrt{-5}]/(2) \longrightarrow \mathbb{Z}[\sqrt{-5}]/(2, 1+\sqrt{-5})$$ $(2)$ and $(2, 1+\sqrt{-5})$ denote ideals. I am told that ...
• 1,345
43 views

### Find all prime ideals in $K[X_1,X_2,X_3]$ which contain $(X_1^2-X_2X_3,\ X_1(1-X_3))$

I have to determine $$\sqrt{(X_1^2-X_2X_3,\ X_1(1-X_3))}$$ in $K[X_1,X_2,X_3]$, where $K$ is a field. I'm supposed to use the fact that $\sqrt{I}=\bigcap_{P\in\text{Spec}A,\ I\subseteq P}P$ for any ...
284 views

• 1,530
1 vote
50 views

### A finite ring homomorphism from reals

Is there a finite ring homomorphism $f$ such that $f:\mathbb{R}\rightarrow A$ for some ring $A$? I am not sure where to begin, but I thought perhaps $f:\mathbb{R}\rightarrow \mathbb{R}[x]$ would work ...
• 243
40 views

### Image of prime and maximal ideals?

Suppose we have a homomorphism $\phi: A \rightarrow B$ where $A$ and $B$ are rings. $\phi$ is injective and not necessarily surjective. Is image of prime ideal and maximal ideals in $A$ also prime and ...
49 views

### If $f(a)$ is invertible under a ring homomorphism $f$, is $a$ invertible too? [duplicate]

Suppose $f\colon R\to S$ is a ring homomorphism (and not rng homomorphism, wherein $f(1) = 1$ is not generally true). I can prove that if $a$ is invertible$^1$, then $f(a)$ also is, with its inverse ...
• 1,967