# 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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### Is the localization of a zero-dimensional ring a quotient?

If $R$ is any commutative zero-dimensional ring and $m$ is a maximal ideal, then is $R_m$ always naturally a quotient of $R$? In other words, is the natural map $R\to R_m$ always surjective? I was ...
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### What do the words "descends" and "Induced" mean in the following quoted passage?

The following is taken from pg 4 section 6.1 of the following notes Background $\quad$ We start by observing that a ring homomorphism descends to quotient rings whenever the image of an ideal on the ...
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### Are there counterexamples of "dividing each other implies association" on a commutative ring but not integral domain? [duplicate]

I am reading about "Fraction on Commutative Ring". On the textbook a proposition states that Let $R$ be a domain and $a,b\in R$. If $a\mid b$ and $b\mid a$, then there exists a unit $u$ ...
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### What is wrong in this proof and where is it needed that $R$ is an integral doamin? [duplicate]

I wanted to prove the following theorem myself: Let $R$ be an integral domain and $p \in R$. If $(p)$ is a maximal ideal, Then $p$ is a prime element. My attempt: Since $(p)$ is maximal then there ...
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### Correspondence theorem and Quotient ring isomorphism

The following is taken from Abstract Algebra A Comprehensive Introduction by: Lawerence and Zorzitto. Background The ideal in a commutative ring $R$ generated by eleements $a_1,\dots, a_n$ is denoted ...
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### Question about finding generators of the kernel for a substitution maps in rings.

The following is taken from Abstract Algebra A Comprehensive Introduction by: Lawerence and Zorzitto. Background Exercsie 6: Find generators for the kernel of each of the following substitution maps: ...
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### Conventional notation for gcd and principal ideal in the context of Bezout domain [duplicate]

Background Definition 1: Let $R$ be a commutative ring with identity, $c\in R$ and let $I$ be the set of all multiples of $c$ in $R$, that is, $I=\{rc\mid r\in R\}$. Set $I$ is an ideal and is ...
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### $\gcd(\text{multiples}(a,b))= \text{lcm}(a, b)$?

I am exploring whether the following assertion holds true in integral domains: $\gcd(\text{multiples}(a,b))= \text{lcm}(a, b)$. Let us make this formal below. Consider two elements $a$ and $b$ in an ...
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### Maximal subgroups of multiplicative group of The Real Quaternions, $\textbf{H}(\mathbb{R})$.

The division ring $\textbf{H}(\mathbb{R})$ also known as the real quaternions, as it was described here, is an extention of the complex numbers. My main goal is to find or describe the maximal ...
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### the ideal $〈6〉$ in the ring $(\mathbb Z,+,.)$

I am trying to solve this past exam question: In the ring $(\mathbb Z,+,.)$, the ideal $〈6〉$ is (a) maximal (b) prime (c) strongly prime (d) another answer. Which option is correct? The only theorem I ...
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### Trouble understanding an exercise needing to use $\Bbb{Q}[x]$ is algorithmic.

Background Exercise 17: This exercise shows that factorization in $\Bbb{Q}[x]$ is algorithmic. Let $f(x)\in \mathbb{Z}[x]$ be a polynomial degree $n$. If $f$ is reducible, it has a factor $g$ of ...
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### Meaning of the notation $\underline{x}$ for denoting coset.

The following is taken from Algebra: Notes from the underground by: Aluffi, Paolo Background Example 6.21 Consider the ring $R=\frac{\Bbb{C}[x,y]}{(y^2-x^3)}$. One can check that $R$ is an integral ...
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### problem on prime ideal [closed]

Let (R,+,∗) be a ring and let (P,+,∗) be an ideal with P≠{0} (0 being the identity for +) and P≠R. Then (P,+,∗) is a Prime Ideal if for all a∗b∈P we have that a∈P or b∈P. Prove it.
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### Concerning some quadratic field extension $[L:\mathbb{C}(u,v)]=2$

In the following question I am actually asking about the answer to this MO question. First, I will ask it generally, then I will present the question and answer appearing in MO. A general question: ...
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### For a commutative ring $R$, is there an algebraic structure corresponding to $R/R^{\times}$, and does it have a meaning? [duplicate]

Question 1. For a commutative ring $R$, if we view $R$ as a commutative monoid, is there an algebraic structure corresponding to $R/R^{\times}$, and does it have a meaning? In detail, is $R/R^{\times}$...
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