# Tag Info

### different definitions of Hopf algebras

This question has been successfully answered if you combine the other answers and the comment discussion that they contain. I will summarize: There are various differences between the definitions. ...
• 3,039
Accepted

### Question on proof that all subgroups of $\mathbb{Z}$ and $\mathbb{Z_n}$ are subrings (and ideals).

Let's prove that every subgroup $A$ of $\mathbb{Z}$ is an ideal. We need to show: $0 \in A$ $\forall a \in A: -a \in A$ $\forall a,b \in A: a+b \in A$ $\forall n \in \mathbb{Z}, a \in A: na \in A$ ...
1 vote

### Question on proof that all subgroups of $\mathbb{Z}$ and $\mathbb{Z_n}$ are subrings (and ideals).

Let $G$ be a subgroup of $\mathbb{Z}$. Let $a,b\in G$, then $a+b,-a\in G$ because it is a subgroup. $ab=a+...+a\in G$ for the same reason. This makes $G$ a sub-rng (ring without $1$) of $\mathbb{Z}$ - ...
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### Exercise about Ideals, Generators, principal ideals and prime ideals

I'll start with correct proofs of each of the claims (assuming I didn't make a mistake), and then comment on each section of your question. I'm putting my answers first because that's the order in ...
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### Exercise about Ideals, Generators, principal ideals and prime ideals

You seem to be using $M$, $A$, and $I$ for the same set here. Stick with $A$ as given in the original problem. Edit Now I see that $M$ stands for something different; a set of generators. Your answer ...
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1 vote
Accepted

### Prove this is an exact sequence

If one of $a,b$ is 0, then the proof is easy. So assume $a,b\not=0$. The last map in the sequence should be $$f: ao + bo \longrightarrow o/(ao:bo)$$ $$as + br \mapsto r$$ To see $f$ is well-defined, ...
• 310
1 vote
Accepted

### Proof that gaussian integers are an integral domain by contradiction.

Although it is already too late, maybe there is a better way to see why your proof is wrong. You see, one of the reasons we want to show something is an integral domain is that cancellation property ...
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1 vote

### Let $f:A→B$ be a ring morphism and $φ : \operatorname{Spec}B→\operatorname{Spec}A$ its induced continuous map, then prove that $\rm im\ φ ⊆ V(\ker f)$

I came up with this solution: Let $P\in \operatorname{Spec}(B)$ lets see that $\phi(P)\in V(\ker (f)) \:$, as $P$ is an ideal we have that $0 \in P\:$ so $f^{-1}(0)\subset f^{-1}(P) \:$ that means ...
1 vote

### Let $f:A→B$ be a ring morphism and $φ : \operatorname{Spec}B→\operatorname{Spec}A$ its induced continuous map, then prove that $\rm im\ φ ⊆ V(\ker f)$

Strong hint: This is essentially the correspondence theorem but for rings rather than groups. Namely, ideals in $A$ that contain $\ker f$ are in correspondence with ideals in $B$, under the same ...
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### If $R$ is a Noetherian ring, then $R^n$ is Noetherian

If $R$ is noetherian, the polynomial ring $R[x_1,\dotsc,x_n]$ is also noetherian. Consider the ideals $$I_k = (x_1, x_2, \dotsc, x_k-1, \dotsc, x_n).$$ The quotients $R[x_1,\dotsc, x_n]/I_k$ are ...
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Accepted

### Trouble with 'elementary questions' as a beginner math student (Soft Question)

Rather than trying to give general suggestions (for this, I recommend Polya's How to Solve it), I'll go through your first example in a way that illustrates the kind of insights you (ideally) should ...
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### Adjoining a function to a ring: what is this called?

Interesting question. Let's use $R \mapsto R\{f\}$ to denote this construction. Note that there is a function $R\{f\} \to R\{f\}$ defined by $r \mapsto f(r)$. By abuse of notation, we'll also call ...
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1 vote
Accepted

### How to show $(k[x,y,z]/(xz,yz))_z\cong k[z]_z$?

\begin{align}\left(k[x,y,z]/(xz,yz)\right)_z&\cong k[x,y,z,t]/(xz,yz,1-zt)\\&\cong k[x,y,z,t]/(x,y,1-zt)\\&\cong k[z,t]/(x,y,1-zt)\\&\cong \left(k[z]\right)_z.\end{align}
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