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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Finding the Kernel and Image of $\mathbb Z \to \mathbb Z[i]/(1+3i)$, $x\mapsto x+(1+3i)$

I'm trying to apply the homomorphism theorem to the following function: $$h:\mathbb Z \to \mathbb Z[i]/(1+3i)$$ $$x\mapsto x+(1+3i)$$ Where $(1+3i)$ is the ideal generated by $1+3i$. I know that ...
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All maximal chains have the same length in $k[x_1,\dots,x_n]$

Let $k$ be a field, and let $A$ be a $k$-algebra of finite type that is an integral domain. Show that any maximal ascending chain of prime ideals in $A$ has length equal to $\operatorname {dim}A$. ...
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Clarify on Krull dimension and integrality

Let $f:A\to B$ be a homomorphism of rings, and let $I:=\operatorname{ker} f$. If $f$ is surjective, $\operatorname{dim}A\ge\operatorname{dim} B$. I'd say that this holds because a chain of prime ...
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Direct Proof that Artinian Rings have Stable Range 1

Is there a direct proof that any (right) Artinian ring has stable range 1? More precisely, let $R$ be a right Artinian ring and $a,b\in R$ be such that $aR+bR=R$. Can we prove that $(a-bt)R=R$ for ...
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If a,b are nilpotent elememts of a ring, then is (a*b) is nilpotent? [duplicate]

I had this in an exam and I cannot prove this as it seems that the ring must be conmutative, any hints?
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bitmasking polynomial $\mathbb{Z}_2[X]$ by multiplying with another one

Suppose I have a polynomial $\mathbb{Z}_2[X]$. That is, this polynomial represents a number in binary form (with just 1 and 0). Let's say that I want to extract the second bit by multiplying by ...
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Exercise with integral extensions

Let $R$ be an integral domain, and let $K$ be its field of fractions. Let $K\subseteq L$ a field extension. Is it true that, if $x\in L$ is integral over $R$, all the coefficients of the minimal ...
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Non-trivial zero divisors of Polynomial Quotient Ring

I'm writing below some things i found in an exercise: Let $f(x)=x^3+x^2+x+1$ and $B:=\mathbb{Z}_2[x]/(x^3+x^2+x+1)$ Since the degree of $f(x)$ is $3$ and I found an evident root $f(1)=0$, then $f(x)$ ...
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Exercise 3.1.2, Bosch's Algebraic Geometry and Commutative Algebra

Let $R$ be a ring and $\Gamma$ a finite group of automorphisms of $R$. Show that $R$ is an integral extension of the fixed ring $R_{\Gamma}:= \{a \in R : \gamma(a) = a\ \forall \gamma\in\gamma\}$. It ...
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Ring of invariants preserves normality

Suppose we have a normal domain $R$ (i.e. integrally closed in its field of fractions) with a group $G$ acting on it by ring homomorphisms. I was wondering how one could prove that the ring of $G$-...
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Are euclidean rings the minimal condition for the greatest common divisor to exist?

In a textbook I read a proof that the greatest common divisor always exist and is unique for euclidean rings. Is there a more general algebraic definition for which the GCD always exists, or are ...
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Can the Chinese Remainder Theorem be proved for a commutative ring without unity? Or if $I+J \neq R$?

I apologize for this not being a more specific question. I am just wondering what conditions are "necessary" for the Chinese Remainder Theorem to hold true. I know that we need a surjective ...
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How called and what are the properties of rings in which there are elements such that division by them is sometimes zero? [closed]

Suppose there is a commutative ring in which there are "infinite" elements, of different infiniteness order, such that division of a finite element or infinite element of small order by an ...
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Question about universal derivation $\Omega_{A/k}$
Let $k$ be a ring, let $A$ be a $k$-algebra. The universal derivation $\Omega_{A/k}$ is the (unique) $k$-module representing the functor of the $k$-derivations of $A$; suppose that $\Omega_{A/k}=0$. ...