# 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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### Does $A^2 \cong B^2$ imply $A \cong B$ for rings?

If $A$ and $B$ are two unital rings such that $A \times A \cong B \times B$, as rings, does it follows that $A$ and $B$ are isomorphic (as rings)? I believe that the answer is no, but I can't come ...
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### $R$ be a Noetherian domain , $t\in R$ be a non-zero , non-unit element , then is it true that $\cap_{n \ge 1} t^nR=\{0\}$?

Let $R$ be a Noetherian domain, $t\in R$ be a non-zero, non-unit element, then is it true that $$\bigcap_{n \ge 1} t^nR=\{0\} \text{?}$$ It almost feels like the nilradical (which is zero for any ...
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### Minimal ideal in a ring which is generated by an idempotent element.

Let $R$ be a commutative ring with unity and $M$ be a minimal ideal of $R$ such that $M = Re$ where $e$ is an idempotent element in $R$. Then $R = Re \oplus R(1-e)$ I am not able to see, in order ...
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### $R$ be an integral domain , $x \in R$ , $I$ an ideal such that $I+\langle x \rangle , (I:x)$ are principal ideals , then is $I$ a principal ideal?

Let $R$ be an integral domain , $x \in R$ , $I$ be an ideal such that $I+\langle x \rangle$ and $(I:x):=\{r \in R : rx \in I\}$ both are principal ideals , then is $I$ also a principal ideal ?
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### Algebra of Endomorphisms of a functor $\mathcal{F}$

Recently, I was reading a paper and stumbled upon something for which I don't find the formal definition unfortunately in any textbook that have in my hands. It's the notion of the endomorphism ...
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### $\Bbb Z[\sqrt{-5}]$ is not a PID [duplicate]

I want to show: In a PID $R$ two elements $a,b\in R$ always have a greatest common divisor. Therefore $\Bbb Z[\sqrt{-5}]$ is not a PID. For the first part: $I=\{ax+by:x,y\in R\}$ is an ideal, ...
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### What are the conditions needed for two principal ideals of a ring to be isomorphic?

Given a commutative ring $R$, and $p(x),q(x) \in R[x]$ monic polynomials, under what conditions on $p(x)$ and $q(x)$ are the principal ideals $\langle p(x) \rangle$ and $\langle q(x) \rangle$ ...
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### Is $\Bbb Z[i]$ a Euclidean ring? [duplicate]

Is $\Bbb Z[i]$ a Euclidean ring? If not, what would be the simplest way of seeing that $\Bbb Z[i]$ is a PID?
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### Given a commutative ring $R$ and a monic polynomial $p(x) \in R[x]$ is $R[x]/\langle p(x) \rangle$ always a finite integral extension of $R$?

I suspect this to be true based on the fact that $p(x)$ is monic, so it should be the case that $R[x]/\langle p(x) \rangle$ is a finitely generated module over $R$, but I have no good reference for ...
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### Quotient ring of Gaussian integers $\mathbb{Z}[i]/(a+bi)$ when $a$ and $b$ are NOT coprime

The isomorphism $\mathbb{Z}[i]/(a+bi) \cong \Bbb Z/(a^2+b^2)\Bbb Z$ is well-known, when the integers $a$ and $b$ are coprime. But what happens when they are not coprime, say $(a,b)=d>1$? — For ...
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### Definition of a simple ring

I'm reading through Lang's Algebra. Lang defines a simple ring as a semisimple ring that has only one isomorphism class of simple left ideals. On the other side, Wikipedia says that a simple ring is ...
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### Left- and right-sided principal ideals have same index?

One fact about the Lipschitz integers (quaternions of the form $a + bi + cj + dk$ where $a, b, c, d$ are integers) is that the left-sided ideal generated by any element $Q$ has the same index in the ...
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### Why is the Rees Algebra Noetherian if the underlying ring is?

Let $R$ be a commutative ring with $1$, $I \subset R$ a proper ideal. The Rees Algebra, with respect to $I$, is defined: $R[It]= \bigoplus_{n=0}^\infty I^nt^n \subseteq R[t]$. In many places I've read ...
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### Extension of a finite field to a finite non commutative ring

Can a finite field be extended to non-commutative finite rings so that not all elements of the field commutes with the elements of the ring? I have been trying this taking the examples of matrices.
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### Why does this ring have rank $k!$?

Let $R$ be any ring free of rank $k$ over $\mathbb{Z}$ having non-zero discriminant. Let $R^{\otimes k} = R \otimes_{\mathbb{Z}} \cdots \otimes_{\mathbb{Z}} R$. Then $R^{\otimes k}$ is a ring of rank ...
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### Advantages and disadvantages of a particular definition of rings and subrings

My professor defines a ring as a triple $(A, +,\ \cdot)$ such that $(A, +, 0)$ is an abelian group, $(A,\ \cdot)$ is a semigroup and $\cdot$ distributes over $+$. Subsequently, $B\subseteq A$ is said ...
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### $f:A\to B$ flat homomorphism of rings, $Q$ is a prime ideal of $B,$ and $P=Q^c.$ Why is $B_Q$ a local ring of $B_P?$

I have a question about Exercise 2.18 on Introduction to Commutative Algebra by Atiyah and MacDonald. Let $f:A\to B$ be a flat homomorphism of rings, let $Q$ be a prime ideal of $B$ and let $P=Q^c,$ ...
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### Showing that two quotient rings are isomorphic

Is $\mathbb{Q}[x]/(x^2-x-1)$ isomorphic to $\mathbb{Q}[x]/(x^2-5)?$ My guess is yes. I am trying to find an isomorphism between the two. Universal Property of Quotient certainly helps. I am thinking ...
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### does algebra over algebraically closed field has isomorphism classes of irreducible modules?

Let $F$ be an algebraically closed field and $M$ be an F-algebra. Which are the conditions that make $M$ has finitely many isomorphism classes of irreducible $M$-modules?
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### What does “coefficients from all of $\mathbf{F} _q$” mean

I was reading Wikipedia's page on Ring Learning with Errors, and came to wonder what is meant by "with coefficients from all of $\mathbf{F} _q$" which is a requirement for the set of known polynomials....
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### Atiyah–Macdonald exercise 14 chapter 1

So here is the part of exercise 14 of chapter 1 that has been bothering me: Let $A$ be a commutaive ring with identity. Let $\Sigma$ be the set of ideals with the property that every element in them ...