# 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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### Flat, non-faithfully flat, submodule of a faithfully flat module

Let $R$ be a unital ring (not necessarily commutative) and let $M$ be a (left) faithfully flat module over $R$. Can there exist a non-zero flat (left) $R$-submodule of $M$ that is not faithfully flat.
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### $u^{2}-1$ is a unit in a ring $A$

Hello I have the next question that I want to prove (or one counterexample if there is one) Let $A$ be a ring with maximal ideal $M$ and quotient field $k=A/M$ of size at least $4$ such that the ...
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### Maximal ideals in certain affine algebras

Let $a_1,\ldots,a_m \in \mathbb{C}[x_1,\ldots,x_n]$, with $m > n$, and let $R=\mathbb{C}[a_1,\ldots,a_m]$. For example, $R=\mathbb{C}[x^2,x^3] \subseteq \mathbb{C}[x]$. Question. Could we find all ...
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### What exactly is a cokernel? What's the motivation behind that, and what's its use? And why is it a quotient module?

Having a homomorphism of $R$-modules $f\colon M \rightarrow N$, we say a cokernel is $N/(\operatorname{im} f)$ which is a $R$-module (quotient). Because $\operatorname{im} f\subset N$, this would mean ...
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### Let $R$ be an arbitrary ring, $I\subset R$ an ideal. Why is $R/I$ an R-module?

We've defined that for any arbitrary ring $R$ and ideal $I\subset R$, $R/I$ is a module and is called a quotient R-module. But why is $R/I$ an R-module in the first place? I can't find anything on the ...
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### Showing $x \in R$ is invertible if and only if $x$ is not contained in any maximal Ideal [duplicate]

Here $R$ is commutative. $\Rightarrow$ is fairly easy to prove as every Ideal containing a unit must be the entire Ring so it cannot be maximal. But I'm having some trouble with $\Leftarrow$, this is ...
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### Prime Ideals in finite direct product of commutative Rings [duplicate]

Let $A_1, \dots , A_n$ be commutative unitary Rings and $A = \prod_{i=1}^{n} A_i$. Then every prime Ideal $\frak{p}$ $\subset A$ is of the form $\pi_i^{-1}(\frak{p}_i)$ where $\pi_i: A \to A_i$ are ...
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### Copies of $\mathbb{C}$ in Artin-Wedderburn decomposition

Let $G$ be a finite group, and let $R = \mathbb{C}G$ be the group algebra. Show that the number of distinct group homomorphisms from $G$ to $\mathbb{C^*}$ equals the number of copies of $\mathbb{C}$ ...
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### Define ring and field

An algebraic ring and field are defined by two inner binary operations. I am only unsure about one thing: do the operators have to be plus and times or are they arbitrary? Almost every book (and ...
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### Are prime ideals mapped to prime ideals?

Let $A$ and $B$ be two commutative rings and $\phi$ be homomorphism between them , if $P$ is a prime ideal of $A$, then is $\phi(P)$ a prime ideal of $B$? I think the statement is false, but ...
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### Let $R$ be a complete local ring with respect to maximal ideal $I$. Then, intersection of $I^n$ is zero.
Let $R$ be a complete local ring with respect to maximal ideal I. I would like to prove intersection of $I^n$, $n\ge1$ is zero. My attempt : $R$ is complete with respect to $I$, so $R$ is Hausdorff ...