# Questions tagged [ring-theory]

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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### Let R be a ring with unit element, R not necessarily commutative, such that the only right-ideals of R are (0) and R. Prove that R is a division ring.

Let $R$ be a ring with unit element, $R$ not necessarily commutative, such that the only right-ideals of R are $(0)$ and $R.$ Prove that R is a division ring. This was a problem from the book, "...
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### Showing that $k[x]$ is integral over $k[x^2-1]$

For context I am working on Atiyah-Macdonald 5.4. I want to show that the extension $k[x^2-1]\subset k[x]$ is integral. I believe this is the case using that $k[x^2-1]=k[x^2]$, which I believe can be ...
1 vote
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### Error in Herstein's "Topics in Algebra", zero ring has characteristic $1$ which is not prime

I was trying to prove the following statement: "If an integral domain has a finite characteristic then the characteristic of the integral domain is a prime number" This made me look at the ...
1 vote
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### Why showing that $\bar{xy} \in \bar{P}^{2}$ but that no power of $\bar{y}$ lies in $\bar{P}^2$ shows that it is a prime ideal?

Here is the question I am trying to understand its solution: Let $R = \mathbb Q[x,y,z]$ and let bars denote passage to $\mathbb Q[x,y,z] / (xy - z^2).$ Prove that $\overline{P} = (\bar{x}, \bar{z})$ ...
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### Difference between $R/P$ and $R_P$

Let $R$ be a commutative ring and $P$ be a prime ideal. Consider the quotient Ring $R/P$ and localization $R_P$, in the Borcherd's lecture, he said $R/P$ making $P$ minimal, but $R_P$ making $P$ ...
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### Enveloping algebra of an algebra essentially of finite type (Weibel 9.4.5)

Let $k$ be a commutative Noetherian ring and $R$ be an algebra essentially of finite type (that is, $R$ is a commutative $k$-algebra and it is a localization of a finitely generated $k$-algebra). In ...
1 vote
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### Affineness of the algebra of formal power series?

Let $\mathbb{k}$ be a field and $A$ be a commutative $\mathbb{k}$-algebra. It is clear that when $A$ is an affine $\mathbb{k}$-algebra (that is, $A$ is finitely generated as a $\mathbb{k}$-algebra), ...
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### The uniqueness of the identity elements are a request in field axioms? [duplicate]

I'm studying analysis and notice a conflict when some authors write about them. Sometimes is uniqueness of the neutral elements are in the axioms, sometimes is a corollary of the axioms which talk ...
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### How to verify that the modulo map is a ring homomorphism?

The map f from $$\mathbb{Z} \to \mathbb{Z}/3\mathbb{Z}$$ that maps every integer to its modulo with 3 is a ring homomorphism. However, I'm having trouble verifying this as if we apply this map to the ...
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### Length of a quotient of $1$-dimensional Noetherian local ring by a regular element

Let $(R,m)$ be a $1$-dimensional Noetherian local ring. $x$ is a nonzero divisor of $R$. How to deduce the following identity $$l(R/(x))=\sum_{P\text{ a minimal prime ideal}}l(R_P)l(R/((x)+P))\ ?$$ ...
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### One sided ideals of a semisimple ring.

Let $A$ be a semisimple ring. I'm wondering whether all ideals of $A$ are two sided. I know that all semisimple rings are both left and right semisimple. And, since $A$ is a semisimple module over ...
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