# 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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### Prime elements of ring $\mathbb{Z}[\sqrt{-21}]$ [closed]

Find prime elements of the ring $\mathbb{Z}[\sqrt {-21}]$. Please help with some ideas.
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### About definition of UFD

On Wikipedia, UFD is defined as an integral domain in which every element can be uniquely factored as product of primes (irreducibles), up to multiplication by units and arrangement. My question is ...
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### When is the norm of a number even?

In the ring $$\textbf{Z}[i],$$ if the norm of an element is divisible by $2$, then the element must be divisible by $$1 + i,$$ and vice versa. A similar result holds for $$\textbf{Z}[\sqrt 3]$$ and ...
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### Is every element of a finite dimensional commutative non-unital $\mathbb{R}$-algebra nilpotent?

Considering a few examples of finite dimensional non-unital algebras over the reals, I tried coming up with an example of such an algebra with non-nilpotent zero divisor elements. In all the examples ...
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### On graded Artinian Gorenstein algebras

Let $k$ be a field and $R$ an $\mathbb{N}$-graded $k$-algebra that is graded-commutative. Assume that $\dim_k R<\infty$ and that $R$ is Gorenstein (i.e. the injective dimension of $R$ over itself ...
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### Show that $R = \mathbb{Z}[\sqrt{-17}]$ is not a Euclidean ring.

Show that $R = \mathbb{Z}[\sqrt{-17}]$ is not a Euclidean ring. To do this I tried showing that the ring is not a principal ideal domain. I wonder if this is enough and how to actually show that it is ...
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### Help with understanding quotient ring structure

Let $R$ be the ring $\mathbb{Z}[x]/((x^2+x+1)(x^3+x+1))$ and $I$ be the ideal generated by $2$ in $R$. What is the cardinality of the ring $R/I$? I am having a hard time understanding what the ring ...
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### Definition of irreducible element of a ring

I found in my notes the following definition: Let $r\neq 0$, $r$ non-invertible. $r\in R$ is called irreducible iff $r=a\cdot b$ with $a,b\in R$ then either $a\in U(R)$ or $b\in U(R)$. Why does ...
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### Singular ideal containing a given nilpotent ideal

Let $R$ be a ring with identity, and $Z(R_R)$ be the singular ideal. Is it true that any nilpotent ideal of $R$ lies in $Z(R_R)$? It is well known that any central nilpotent element would belong to ...
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### Not fully understanding polynomial quotient rings.

This is my (informal) understanding of a quotient ring. I understand that this is very flimsy, but I hope you can get the main idea. You have some ring $R$ and you want to quotient out an ideal $I$...
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### If $R$ is an integral domain and $R[x]$ is an euclidean domain, then $R$ is a field [closed]

Is this obvious? I cannot see that this is true. The converse is fairly obvious though. I tried to show $(x)$ is a maximal ideal and try the quotient but failed. I will appreciate any help.
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### If $A$ is an integral domain with a finite number of primes then $Q(A)=A_a$ for some $a \in A$ [closed]

If $A$ is an integral domain with a finite number of prime ideals is it possible to get the field of fractions localizing only by a set $\{a^k\}$?
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### Finding units in quadratic integer rings

I want to find the units in $\mathbb{Z}[\alpha]$, where $\alpha=\frac{1+\sqrt{-11}}{2}$. One can of course use norms to find the units in quadratic integer rings of the form $\mathbb{Z}[\sqrt{D}]$ ...
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### Restricting the quotient map of rings to a subring

When $q$ maps $R$ to $R/I$ and $p$ is the restriction of $q$ to a subring $A$ of $R$, why is the image of $p$ $(A+I)/I$? $q$ maps $r$ to $r+I$, so shouldn't $p$ map $a \in A$ to $a+I$, so image of $p$...
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### Bijection between compact space $K$ and maximal ideals of real-valued functions on $K$

Let $K$ be a compact topological space, and denote by $R$ the ring of continuous functions $K \to \mathbb{R}$, with addition and multiplication defined pointwise. We prove that there is a bijection ...
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### Cardinal of quotient rings of gaussian integers. [duplicate]

It is known that $\mathbb{Z}[i]$ is a PID and that $\mathbb{Z}[i]/(a+bi)\mathbb{Z}[i]$ is finite for all $(a,b) \in \mathbb{Z}^2\backslash \{(0,0)\}$. My question : Is there any result on the ...
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### Converse to Chinese remainder, domain, isomorphism as rings

The converse to CRT asks: does $R/(I \cap J)\simeq R/I \times R/J$ imply $I+J=R$? For me $\simeq$ is an abstract ring isomorphism, sending $1$ to $1$, not necessarily $R$ linear, i.e., one of $R$--...
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### Division ring if and only if it has no proper left ideals.

Let $R$ be an associative ring with $1$. Prove that $R$ is a division ring if and only if $R$ has no proper left ideals. Clearly, if $R$ is a division ring and $I\neq\{0\}$ is a left ideal, then ...
### Any subring of $A$ is an ideal. If $A$ is an integral domain then $A$ is commutative
Any subring of $A$ is an ideal. If $A$ is an integral domain then $A$ is commutative. Is my proof correct? So let $a$ and $b$ nonzero elements of $A$. $C(a)=\{ x\in A \mid ax=xa\}$ Is a subring ...
### Show that $R_P$ has a unique maximal ideal
Problem is: Let $R$ be a commutative ring and let $P$ be a prime ideal. (a) Prove that the set of non-units in $R_{P}$ is the ideal $P_{P}$. (b) Deduce that $R_{P}$ has a unique ...