# Tagged Questions

This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

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### Why the map sending $a\otimes b$ to $(ab, a\bar{b})$ is injective?

I am wondering why the ring homomorphism $\phi : \mathbb{C}\otimes_\mathbb{R}\mathbb{C} \longrightarrow \mathbb{C}\times\mathbb{C}$ sending $a\otimes b$ to $(ab, a\bar{b})$ is one-to-one. So any ...
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### $\mathbb C[x_1,\ldots,x_n]/I=\mathbb C\times\cdots\times\mathbb C$.

Let $A=\mathbb C[x_1,\ldots,x_n]/I$ and for every $y\neq 0$, we have $y^2\neq 0$ and $\dim A=0$. I would like to prove that $A=\mathbb C\times\cdots\times\mathbb C$. Attempt of a solution $\dim A=0$...
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### Intersection of $max(R)$ with a closed subset in $Spec(R)$

Let $R$ be a commutative ring with unity and $E$ be a nonvoid closed subset of $Spec(R)$. If $U$ is an open subset of $Spec(R)$ with $E∩Max(R)⊆U$, where $Max(R)$ is the set of maximal ideals of $R$, ...
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### Jacobson radical of a ring finitely generated over $\mathbb Z$

If a commutative ring $R$ with $1$ is finitely generated over $\mathbb Z$ could one deduce that the Jacobson radical of $R$ is nilpotent? I am aware of the well-known fact that when $R$ is artinian,...
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### Is there an online database somewhere that lists identities for algebraic structures with two binary operators?

I'm working on an abstract algebra library in Python, and I'm trying to include as many functions that analyze algebraic structures, returning true or false based on whether or not the algebra ...
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### Proving that a zero ring cannot have a homomorphism to a unital ring.

Q: Prove that a zero ring cannot have a homomorphism to a unital ring. I don't know how to prove this. If $f:R\to S$ is the homomorphism, and $f(0_R)=0_S$, then I don't see which homomorphism ...
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### Why $\mathbb{C}\otimes_\mathbb{Q}\mathbb{C} \not\cong \mathbb{C}\times\mathbb{C}$

I am wondering if this is true or not: $\mathbb{C}\otimes_\mathbb{Q}\mathbb{C} \cong \mathbb{C}\times\mathbb{C}$ (as rings). Any help or suggestion would be helpful.
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### Number of non trivial ring homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{28}$ [duplicate]

One homomorphism is $1 \mapsto 1$. Other homomorphisms are: We know that if $f$ is a homomorphism from $R$ to $S$ and $f(1_R) \neq 1_S$, then $f(1_R)$ is a zero diviasor in $S$. So zero divisor ...
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### ideal calculations: $2\mathcal{O}_K=\mathfrak{B}^4$ in the ring of integers of $K=\mathbb{Q}(i,\sqrt{2m})$

Let $K=\mathbb{Q}(i,\sqrt{2m})$ where $m \in \mathbb{Z}$ is odd and squarefree. Let $\alpha = (1+i)\sqrt{2m}/2$. Then $\alpha^2=im$, such that $\alpha$ is part of the ring of integers $\mathcal{O}_K$. ...
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### Prove that $B_{q}$ is flat over $B_{p}$
I'm doing this exercise in "Introduction to Commutative Algebra" of Atiyah and get confused by the hint in this book. Here is the exercise: Let $f: A \rightarrow B$ be a flat homomorphism of ...