# Tagged Questions

27 views

### Maximal and prime ideals of $\mathbb{Z} \times \mathbb{Z}$

I have to find a maximal ideal of $\mathbb{Z} \times \mathbb{Z}$ , and a prime ideal that is NOT maximal. Or, essentially, I want $I$ such that $\mathbb{Z} \times \mathbb{Z} / I$ is a field, and I ...
11 views

### How does one find a minimal primary decomposition?

What exactly does it mean for a primary decomposition to be "minimal" and is the a general method to obtain such decompositions? I've tried looking at some examples but they all give very little ...
26 views

### Common divisors in a PID

Suppose that R is an integral domain and that α, β, γ ∈ R. We say that γ is a common divisor of α and β if γ|α in R and γ|β in R. Suppose that R is a PID. Suppose that α, β ∈ R. Let I = (α) and ...
102 views

### Prove: The pre-image of an ideal is an ideal.

Let $\phi : R \to S$ be a homomorphism. If $N$ is an ideal of $S$, then $\phi ^{-1} (N)$ is an ideal of $R$.
31 views

### Principal prime ideal is generated by irreducible element

$R$ is an integral domain, $x\in R$ and $(x)=I$ is a prime ideal. Prove that $x$ is an irreducible element of $R$. So I assume $ab\in I$, with $a, b \in R$. Since $I$ is a prime ideal, either $a$ ...
42 views

80 views

128 views

### Ideal of ideal needs not to be an ideal

Suppose I is an ideal of a ring R and J is an ideal of I, is there any counter example showing J need not to be an ideal of R? The hint given in the book is to consider polynomial ring with ...
115 views

### Find ideals of ring

I am stuck with a homework problem. Let $R=\mathbb{Z}[\sqrt{ -3}]$. a) Find an ideal $I$ of $R$ such that $(4) \subsetneq I \subsetneq R$. Explain why the inclusions $\subsetneq$ in my example are ...
179 views

I have to prove that every finite ring is Noetherian. I know examples of Noetherian rings which are not finite such as the field of complex numbers or a PIR like the integers. But anyway: [Proof]: I ...
174 views

### Product of ideals corresponding to vanishing of points is equal to their intersection

Let $k$ be some field, and let $v,v',v''$ be three distinct points in $k^3$. Let $\mathfrak{m}_v = (X_1 - v_1,X_2 - v_2,X_3 - v_3)$ be the ideal in $k[X_1,X_2,X_3]$ corresponding to the polynomials ...
167 views

### Finding the ideals in a ring of fractions

I am dealing with the ring $$R=\left\{\frac{a}{b} \mid a,b\in\mathbb{Z}\mbox{, b is not divisible by 3}\right\}$$ with addition and multiplication as defined in $\mathbb{Q}$ and I'm trying to find ...
45 views

139 views

### What are the prime ideals of $\mathbb{R}[x_1,x_2,x_3,…]$

What are the prime ideals of $\mathbb{R}[x_1,x_2,x_3,...]$? (this is the ring of polynomials over the reals with countably infinite many indeteminates). My attempt: I think taking the principal ...
1k views

### A proof that this set is an ideal of a commutative ring

This is a homework problem which I have worked hard on, but got stuck at the last step. Any assistance would be much appreciated. The problem is from Herstein's Abstract Algebra, 3rd ed., section 4.3, ...
This is homework from videolecture: "Show that $(x^2-y)$ is prime but not maximal in $C[x,y]$". Linked SE pages offer to approach this by demonstrating that $C[x,y]/(x^2-y)$ is integral domain but not ...