# Questions tagged [ring-homomorphism]

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

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### Is there a non-constant function $f$ such that $f(n) - 1 = 0$ for all $n \in \mathbb{N}$?

So I've been given rings $R = Map(\mathbb{R},\mathbb{R})$ and $S = (a_{n})_{n \geq 0}$ such that $a_{n} \in \mathbb{R}$ and the ring homomorphism $$\phi: R \rightarrow S\\ f \rightarrow (f(n))_{n}$$ ...
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### First isomorphism theorem with sequences

I have the following here: Let $R=\text{Map}(\mathbb{R},\mathbb{R})$ with addition and multiplication defined, as usual, by $(f+g)(x)=f(x)+g(x)$ and $(f \cdot g)(x)=f(x)g(x)$. Let $S$ be the ring of ...
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### How to show this ring homomorphism is surjective?

So I've been given the following R = Map(ℝ, ℝ) with addition and multiplication defined by $(f +g)(x) = f(x)+g(x)$ and $(f ·g)(x) = f(x)g(x)$. Let S be the ring of sequences $(a_{n})_{n}≥0$ with ...
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### Ring homomorphism $\varphi : m\mathbb Z \to \mathbb Z _{\frac{n}{m}}$

This is what I have to prove: The map $\varphi : m\mathbb Z \to \mathbb Z _{\frac{n}{m}}$ where $m \mid n$ given by $\varphi(x) =\overline{\left(\frac{x}{m}\right)}$ is a (not necessarily unital) ring ...
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### Is the map $R^\text{op} \to \text{End}_R(M)$ defined by $a \mapsto (x \mapsto ax)$ valid?

Let $R$ be a not-necessarily commutative ring, and suppose $M$ is left $R$-module. Let $R^\text{op}$ consist of elements written as $a^\text{op}$, where as an element $a^\text{op} = a$ for all $a\in R$...
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### Does an $R$ algebra always contain $R$?

In an associative algebra with unit over a commutative ring $R$ it's true that $R$ is inside the algebra? And, is $1$ is the unit in the algebra, is this inclusion $R\cdot1$?
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### Show there exists an injective ring homomorphism that makes the diagram commute.

I am taking some practice exams to relearn algebra. I'm having trouble with questions regarding showing there exists or doesn't exist a ring homomorphism. Here's a particularly scary-looking one I can'...
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### Ring homomorphisms between convolution rings $R_{2^n}$ and $R_{2^k}$ for $k=1,2,\ldots,n-1$.

I'm currently studying weaknesses in the algebraic structure of the convolution rings used in the NTRU cryptosystem. Moving forward, let $R_q:=(\mathbb{Z}/q\mathbb{Z})[X]/\langle X^N-1\rangle$ with $N$...
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### Quotients of multivariate polynomial rings - is $k[x][y]/(y-x^2) \cong k[y][x]/(y-x^2)$?

This question is motivated by exercise 1.1 in Hartshorne Algebraic Geometry. One has to prove that $k[x,y]/(y-x^2)$ is isomorphic to a polynomial ring in one variable. I know that this can be done by ...
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### Determine the ring homomorphism with domain $\mathbb{Q}?$ [duplicate]

Determine all the ring homomorphism from $\mathbb{Z}$ to $\mathbb{Q}$ and $\mathbb{Q}$ to $\mathbb{Z}$ My attempt : Ring homomorphism from $\mathbb{Z}$ to $\mathbb{Q}$ Here $\mathbb{Z}$ is cyclic ...
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### Are $\mathbb{Q}(π)$ and $\mathbb{Q}(π^2)$ the same field?

I've tried to check if $\mathbb{Q}(π)$ and $\mathbb{Q}(π^2)$ are equal fields. I know there exists a field isomorphism between $\mathbb{Q}(π)$ and $\mathbb{Q}(π^2)$ since $π$ and $π^2$ are ...
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### Does every evaluation map $g: \Bbb{Z}[X,Y]/(\ker \pi) \to \Bbb{Z}$ factor some evaluation map $f:\Bbb{Z}[X,Y] \to \Bbb{Z}$?

Let $\Bbb{Z}[X,Y]$ be the polynomial ring. I know that every evaluation ring hom $f: \Bbb{Z}[X,Y] \to \Bbb{Z}$ is determined by where you send $X$ and $Y$. Let $I = (X^2 - Y^3)$ for example, but it ...
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### The homomorphic images of ring centers are central (Dummit and Foote Exercise 7.3.16 and related question)

Let $\phi: R \rightarrow S$ be a surjective homomorphism of rings. Prove that the image of the center of $R$ is contained in the center of $S$. Which is from Dummit and Foote Exercise 7.3.16. I have ...
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### On an Isomorphism of Semigroup Rings via Congruence Classes

Let $\mathbb Z_{\geq 0}$ denote the set of non-negative integers. Let $\mathbb Z_{\geq 0}^n$ denote the set of $n$-tuples of non-negative integers. (Theorem 2.1.5, Herzog, 1969) Given a finitely ...
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### A surjective homomorphism which is not an isomorphism for rings (Using basic algebra)

Give an example of a ring $R$ and a surjective homomorphism $R \to R$ that is not an isomorphism. Here is my approach. I know a lemma that says that if $R$ is a Noetherian ring, and $f: R \to R$ is a ...
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### Verify $f: \mathbb{Z} ⟶ \mathbb{Z}_{r} \times \mathbb{Z}_{s}$ given by $a ⟶ (ā_r, ā_s)$ is an epimorphism

Let $r,s \in \mathbb{Z}^+$ such that $(r,s)=1.$ Consider the ring $\mathbb{Z}_{r} \times \mathbb{Z}_{s}$ (with respect to ordinary multiplication and addition, component by component), prove that the ...
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### Example to show that $\mathbb Z^3$ is not a quotient of $\mathbb Z[X]$

I am asked to prove that $\mathbb{Z}^n$ is not a quotient of $\mathbb{Z}[X]$ for any integer $n \geq 3$ by showing that there is no surjective ring homomorphism $\mathbb{Z}[X] \to \mathbb{Z}^n.$ We ...
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### Let $M = K + L$ and let $f: M \to N$ be an epimorphism. Prove that $N = f(K) \oplus f(L)$ if $K \cap L = \operatorname{Ker}f$

Let $M = K + L$ and let $f: M \to N$ be an epimorphism. Prove that $N = f(K) \oplus f(L)$ if $K \cap L = \operatorname{Ker}f$ I think we should start with take an element from $K + L$ and then we can ...
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### How to count the number of selected element in each slot within the packed ciphertext

Given an encrypted ciphertext (n slots, packed n elements into a single ciphertext), such as $ct=\{(2,0,1,2),(3,2,1,3),(3,4,0,4),(5,1,4,2)\}$. Formally, $n$ slots can be expressed as $m$ blocks, each ...
Question If $E,F$ are fields and $\beta:F[x]\rightarrow E$ a homomorphism of rings. Show that the kernel of $\beta$ is a maximal ideal or a zero ideal. I just wonder where does the zero ideal case ...
### Why must a ring homomorphism from $\mathbb Q[\sqrt n] \to R$ be the identity mapping?
Let $\mathbb Q[\sqrt d]:= \{a+b\sqrt d \, |\, a,b \in \mathbb Q\}$, and $d\in \mathbb N$ with the condition that $d$ is not a square number. I have shown that this is a subring of $\mathbb R$ and that ...