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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Let $R$ be a ring with identity element such that every ideal of which is idempotent or nilpotent. Is it true that the Jacobson radical $J(R)$ of $R$ is nilpotent? If $R$ is Noetherian and $J(R)$ is ...
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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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Correspondence theorem for rings.

Could someone provide a reference that includes a full and honest proof of the Correspondence Theorem for rings? Let $A$ be a multiplicative ring with identity and $I$ an ideal of $A$. There is a ...
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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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Prove that if $I$ is maximal, then $R[X]$ is a PID. [duplicate]

Let $R$ be a commutative ring with unity such that $R[X]$ is a UFD. Denote the ideal $\langle X\rangle$ by $I$. Prove that If $I$ is maximal, then $R[X]$ is a PID. If $R[X]$ is a Euclidean Domain ...
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$R/Rg$ is a field iff $g\in R$ is irreducible.

Let $R$ be a PID and $g\in R$. I want to show: $R/Rg$ is a field iff $g\in R$ is irreducible. I.e. I want to show that all $a\notin Rg$ are invertible modulo $g$ iff $g$ is irreducible. So if I ...
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Bijection beteween maximal ideals

We know that if $R$ and $I$ an ideal of $R$, then there is a bijection between the prime ideals of $R$ containing $I$ and the prime ideals of $R/I$. It is given by $P\mapsto P/I$. Is it true that this ...
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Upper bound for the no. of maximal ideals of a finite commutative ring ( with unity ) in terms of the order of the ring and/or its number of units?

Is there any known upper bound for the no. of maximal ideals of a finite commutative ring ( with unity ) of order $n$ , in terms of $n$ and /or in terms of the no. of units of the ring ? Say , does ...
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Prove that up-to isomorphism there are two integral domains of order $p^2$.

Prove that up-to isomorphism there is exactly one integral domain of order $p^2$ . Does there exist only two non-commutative rings of order $p^2$ upto isomorphism? We know that any group of ...
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Is a finite inverse limit of noetherian rings noetherian?

Let $\{A_i\}$ be an inverse system of (commutative, unital) Noetherian rings with a finite index set. Is $\varprojlim A_i$ also a Noetherian ring?
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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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Is it possible to endow $\text{GL}_2(\Bbb R)$ with a ring structure?

My question is the following: Is it possible to find a binary operation $*$, seen as an addition, such that $(\text{GL}_2(\Bbb R),*,\cdot)$ has a ring structure (not necessarily with a unit)? [We ...
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$\mathbb{F}$-subalgebra generated by a set

Assume that $A$ is an $\mathbb{F}$-algebra, where by $\mathbb{F}$ I just denote an arbitrary field. Furthermore, if $X \subset A$ is a proper subset, how do we define the $\mathbb{F}$-subalgebra of $A$...
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Trivial extension of an opposite algebra

Suppose that $A$ is a finite dimensional $K$-algebra, where $K$ denote an algebraically closed field. Call $DA=Hom_k(A,k)$. $DA$ admits an $A$-$A$-bimodule structure in the obvious way. The trivial ...
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Annihilator of maximal ideals

Let $R$ be a Noetherian ring. Suppose all the nonzero proper ideals of $R$ have nonzero annihilators. Show that if $M$ is a maximal ideal of $R$ , then $\exists$ $x \in R$ such that $M$ = $ann(x)$ (...
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Letting $\phi:k[x,y]\to k[x]_x$, $\phi(x)=x$, $\phi(y)=\frac{1}{x}$, we see that $\ker \phi$ is prime, and $(1-xy)\subseteq\ker\phi$. Now, given that $k[x,y]$ has Krull dimension 2, $\ker\phi\neq (1-... 0answers 27 views Questions Concerning Proof of Artin-Rees Lemma I have two questions about the proof of the Artin-Rees Lemma presented here:$\textit{Question 1:}$Am I correct in assuming$I^0=R$? This is the only way some of the statements make sense I think. ... 0answers 39 views Advantages and disadvantages of a particular definition of rings and subrings My professor defines a ring as a triple$(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 ... 0answers 20 views $f:A\to B$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$P=Q^c,$... 1answer 13 views If$B$is a commutative domain,$Aut(B)$acts on$Der(B)$by conjugation I'm reading Algebraic Theory of Locally Nilpotent Derivations by Gene Freudenberg, and I don't understand what's meant on the line$Aut(B)$acts on$Der(B)$by conjugation:$\alpha \cdot D = \alpha ...
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I'm studying basic Ring Theory. And in my textbook, the author states the definition of Euclidean domain: The integral domain $R$ is called to be a Euclidean domain precisely when there is a ...
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Showing $\mathbb Z[x]/<5,x^3+x+1>$ is a field

I want to show that $\mathbb Z[x] /<5,x^3+x+1>$ I am currently self-studying and it is a problem of previous graduate school entrance exam. I studied abstract algebra through Fraleigh's book, ...
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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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Mal'cev condition for variety of rings generated by finite fields to be arithmetical

This is an exercise of Burris & Sankappanavar (Universal Algebra), Chapter II, section 12. It asks to prove that, if $V$ is a variety of rings generated by finitely many finite fields, then $V$ is ...
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Two questions about integral “splitting ring” extensions

We have a ring $R$, commutative, and $f_1,\dots,f_n$ polynomials in $R[x]$ monic, with $\deg f_i\ge 1$. It is straightforward to show that there is a ring extension $R\subset S$ such that $S$ contains ...
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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 ...