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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1answer
46 views

Some weaker axiom than “no nontrivial zero divisors.”

I would like to know if there a standard term for or well-known applications of the following axiom for rings or semigroups with zero (which is weaker than the "no nontrivial zero divisors" axiom): ...
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1answer
31 views

Characteristic of Quotient Ring

Consider a ring $R$ with characteristic $n \gt 0$. Let $I$ be an ideal of $R$ with characteristic $m$. I have proved that $m$ divides $n$. Now I am interested about the characteristic of $R/I$. One of ...
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1answer
35 views

Show for any prime $p$ and $a \in \mathbb{F}_{p}$ that $x^p-a$ has multiple roots

Show for any prime $p$ and $a \in \mathbb{F}_{p}$ that $x^p-a$ has multiple roots using the derivative of $x^p-a$ which is $px^{p-1}$ if they are relatively prime then $x^p-a$ only has simple roots. ...
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1answer
19 views

Question about quotients of reducible elements and modules

First, I apologize if this is a stupid question. We started doing modules in my class a few days ago and I'm totally lost with the basics, I think. Suppose we have a ring $K$ and $M$ a $K$-module. ...
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0answers
24 views

Find kernel generators for ring maps

This is the textbook question: Q: Find generators for the kernels of the following maps: $\mathbb{R}[x,y] \to \mathbb{R}$ defined by $f(x,y) \rightsquigarrow f(0,0)$ $\mathbb{R}[x] \to \mathbb{C}$ ...
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4answers
88 views

Question about maximal ideals?

I'm reading Freligh and introduction to Abstract algebra and I'm getting confused. The set generated by $\langle x^2 + 1\rangle$ is a maximal ideal in $R[x]$. First, I don't understand it. $\langle ...
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1answer
49 views

Show the set of units $R^\times$ in $R$ forms a group

Let $R$ be a ring with unity. Show that the set of units $R^\times = \{r ∈ R\mid \exists r^{−1} \in R, rr^{−1} = r^{−1}r = 1\}$ in $R$ forms a group. proof: Since $R$ is a ring, $\exists 1\in ...
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1answer
35 views

subring of a quotient field

Let $R$ be a principal ideal domain, and $S \subseteq Q(R)$ a subring of the quotient field of $R$, so that $R \subseteq S$. I want to show that, for any $x, y \in R$: $$\frac{x}{y} \in S \implies ...
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1answer
34 views

Galois extension of $\mathbb Q$ of degree $n$

I have a basic question in Galois theory. For any given natural number $n$ is there a Galois extension of $\mathbb Q$ of degree $n$? I want to show that there are splitting fields of polynomials in ...
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2answers
58 views

Basic rings (e.g. non commutative) $A$ such $A^n \simeq A^m$ and $n\neq m$

EDIT : precision and broadening of my question. Almost all is in the title : I am looking for various structures $A$ such $A^n$ and $A^m$ (products of $A$) are isomorphic (in the sense that it is ...
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1answer
75 views

Prove there generally is no isomorphism between $R[x]/(x^2-a)$ and $R^2$

I have a ring $\mathbf R =(R, +, -, ., 0, 1)$ (note that there is no invers for multiplication, $R$ is not $\mathbb R$, it is any set for the given algebra). How do you prove that the following does ...
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2answers
34 views

Let $D$ be an integral Domain. Show that $\langle r\rangle = \langle s\rangle$ then $s = ur$ for some unit $u \in D$

My basic question is about the notation- in regards to $\langle r\rangle$ and $\langle s\rangle$. What exactly are these things? I have seen that bracket notation in Group theory. Mainly in relation ...
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0answers
30 views

Rings where action of automorphisms on maximal ideals is transitive

If $R$ is a commutative ring, $\alpha: R \to R$ an automorphism of $R$, and $M$ a maximal ideal of $R$, then $\alpha(M)$ is also a maximal ideal of $R$ with the same quotient field. So the group of ...
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1answer
20 views

For fractional ideal $I$ why is $I\cap R \supsetneq \{0\}$?

In the proof in my textbook that a fractional ideal $I$ in a quotient field $K$ of an integral domain $R$ has an inverse $$I^{-1} = \{ x\in K : x I \subseteq R\}\,,$$ it is used that there exists an ...
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0answers
22 views

Simple examples of fractional ideals

Let $K$ be the quotient field of an integral domain $R$. A fractional ideal $I$ is a subset of $K$ not $\{0\}$, for which a $0 \neq r \in R$ exists so that $r I \subseteq R$ is an ideal in $R$. ...
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2answers
78 views

Annihilators of elements of a finitely generated faithful module over a noetherian reduced ring

Lately I've been thinking to annihilator of modules and I've conjectured a proposition I can't prove, so I'll expose my claim. Let $A$ be a noetherian reduced (commutative) ring and let $M$ be a ...
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1answer
36 views

If $A$ is a Krull ring of $K$, then $A \cap K'$ is also Krull for $K' \subset K$.

If $A$ is a Krull ring of $K$, then $A \cap K'$ is also Krull for $K' \subset K$ (where $K'$ is a subfield of $K$). What's confusing me is that $K'$ may not contain the uniformising element $t$ ...
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2answers
54 views

Definition of a prime

Whilst studying ring theory I came across the following definition of a prime: Given a ring $R$ we say that $r\in R \setminus \{{0}\}$ is prime if $r$ is not a unit, and $r|xy$ implies that $r|x$ or ...
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1answer
46 views

Some ring isomorphic to $\mathbb Z[i] / \langle 5 \rangle$

I am trying to find some ring isomorphic to $\mathbb Z[i] / \langle 5 \rangle$ . I know that $\langle 5 \rangle=\langle (2+i)(2-i)\rangle=\langle 2+i\rangle \langle2-i\rangle$ , now if $d$ is the ...
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2answers
39 views

Question about working in modulo?

This question is in essence asking for understanding of a step in Fermats theorem done Group style. For any field the nonzero elements form a group under field multiplication. So let us take the ...
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1answer
34 views

Hints on how to approach a problem concerning rings/field in Abstract Algebra

I am a student, prepping for a final exam in graduate Abstract Algebra. My professor has told me that he will be giving us the following two problems in class to turn in: (1) Given that R is an ...
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1answer
72 views

Product of principal ideals: $(a)\cdot (b) = (a b)$

In which kinds of rings $R$ does the following hold: $$(a)\cdot (b) = (ab) \; ?$$ With $a, b\in R$, $(a)$ denoting the (two-sided) ideal generated by $a$ and the multiplication of ideals $I, ...
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2answers
51 views

a question about abstract algebra,the order of $\Bbb Z_{5}[x]/ (x^3+x+1)$

Firstly, I have proven that $x^3+x+1$ is irreducible in $\Bbb Z_{5}[x]$,then how can I know the order of $$\Bbb Z_{5}[x]/ (x^3+x+1),$$ where $(x^3+x+1)$ means the ideal generated by $x^3+x+1$. Can ...
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1answer
63 views

Examples of rings whose polynomial rings have large dimension

If $A$ is a commutative ring with unity, then a fact proved in most commutative algebra textbooks is: $$\dim A + 1\leq\dim A[X] \leq 2\dim A + 1$$ Idea of proof: each prime of $A$ in a chain can ...
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2answers
34 views

In a ring $(A,+, \cdot)$ if $aba = a$ then $bab = b$ and all element non zero in $A$ is invertible.

Let $\left(A,+, \cdot\right)$ be a ring with $1$ that satisfies the following condition: For any nonzero $a\in A$, there exists a unique $b\in A$ such that $aba = a$. Show that this $b$ also ...
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0answers
28 views

Splitting field of $f(x)=x^4+3$ in $\mathbb{Q}[x]$

I am trying to find the splitting field of $f(x)=x^4+3$ over $\mathbb Q$. It is irreducible, and the roots are ...
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1answer
38 views

Show that $α$ is a perfect square in the quadratic integers in $\mathbb Q[\sqrt{d}]$

Question: Suppose that $\mathbb Q[\sqrt{d}]$ is a UFD, and $α$ is an integer in $\mathbb Q[\sqrt{d}]$ so that $α$ and $\barα$ have no common factor, but $N(α)$ is a perfect square in $\mathbb Z$. How ...
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1answer
40 views

Does $\text{End}_K(K^n) \cong \text{Mat}(n\times n, K)$?

Let $K$ be a field, $K^n$ a vector space over $K$. Is the following true? $\text{End}_K(K^n) \cong \text{Mat}(n\times n, K)$ Does this change if $K$ is a ring, and $K^n$ a module over $K$?
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21 views

Is there an easy proof to this theorem due to Nagata?

If $D$ is an integral domain satisfying ACCP, let $(p_i)_{i \in I}$ be the family of prime elements in $D$ and $S$ a multiplicative system of $D$ in which every element is the product of prime ...
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2answers
50 views

How to show the following ideal is or isn't maximal?

$I$ is an ideal of $R=\mathbb{Z}[x]$. If $I$ is the set of polynomials $p(x)$ such that $p(3)$ is even. I know that if $R/I$ is a field, then it is maximal. I just don't know how this quotient ring is ...
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1answer
33 views

Are these ring structures on $ (\Bbb{C} \oplus \Bbb{C},+) $ isomorphic?

I have two binary operations, $ \bullet $ and $ \star $, defined on the abelian group $ (\Bbb{C} \oplus \Bbb{C},+) $ by $$ (a,b) \bullet (c,d) = (a c,b d) \quad \text{and} \quad (a,b) \star (c,d) = ...
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0answers
66 views

Prove $λ$ must divide one of $x$, $y$, or $z$.

Let $λ = (3 + \sqrt{−3})/2 \in \Bbb Q[\sqrt{−3}]$. Prove that if $x^3 + y^3 = z^3$, and $x$, $y$, $z$ are quadratic integers in $\mathbb Q[\sqrt{−3}]$, then $λ$ must divide one of $x$, $y$, or $z$. ...
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16 views

Let $I = (x^2, y)$ be an ideal of $\mathbb{Q}[x,y]$ show that Rad$(I)$ = $(x,y)$ and $I$ is a primary ideal that is not a power of a prime ideal

Let $I = (x^2, y)$ be an ideal of $\mathbb{Q}[x,y]$ show that Rad$(I)$ = $(x,y)$ and $I$ is a primary ideal that is not a power of a prime ideal. I can see that $(x,y) \subset Rad(I)$ b/c $x^1, y^2 ...
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1answer
42 views

Power of a sum in a factor ring

Let $n\in$ $\mathbb N$. Find a closed formula for $$(c_1x+c_2y+c_3z)^n$$ in the factor ring $\mathbb Z[x,y,z] / \langle x^2,y^2,z^2\rangle$. What exactly does it mean to find a closed formula? ...
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1answer
19 views

$O_k=\mathbb{Z}[\sqrt{d}]$, whenever $K=\mathbb{Q}(\sqrt{d})$ and $d\neq 1$ mod $4$

I'm going through a proof in my lecture notes for the mentioned statement. Showing $\mathbb{Z}[\sqrt{d}]\subseteq O_K$ was easy to understand, but then there's a few gaps when showing that ...
1
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1answer
21 views

Prove $R$ conatins an ideal that is not finitely generated. $R = F[x,x^2 y,\ldots,x^n y^{n-1},\ldots]$

Prove R conatins an ideal that is not finitely generated. $R = F[x,x^{2}y,\ldots,x^n y^{n-1},\ldots]$ and is a subring of $F[x,y]$ where $F$ is a field. Seems like $R$ itself is not finitely ...
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1answer
19 views

$R$ has no zero divisors . Let $a \in R$ , $0 \ne b \in R$ and $0 \ne n \in \mathbb Z$ be such that $na+ab=0$ , then is it true that $a=0$?

Let $R$ be a commutative ring (not necessarily with unity) with no zero divisors . Let $a \in R$ , $0 \ne b \in R$ and $0 \ne n \in \mathbb Z$ be such that $na+ab=0$ , then is it true that $a=0$ ?
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1answer
40 views

A radical ideal in a commutative ring is prime if and only if it is not an intersection of two radical ideals properly containing it?

Let $I$ be a radical ideal (i.e. $\sqrt I=I$) in a commutative ring with unity. Then is it true that $I$ is a prime ideal if and only if it is not an intersection of two radical ideals properly ...
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1answer
36 views

Prove by counterexample that $\gamma$ and $\delta$ are not necessarily unique

Assume $\mathbb Q[\sqrt{d}]$ is a Euclidean Field and $\alpha$, $\beta$ are two quadratic integers in $\mathbb Q[\sqrt{d}]$, so that there exists integers $\gamma$ and $\delta$ in $\mathbb ...
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3answers
81 views

Let $R$ be a finite ring (with unity) and $S$, $T$ be subrings of $R$. Is $S \cup T$ a subring of R?

Let $R$ be a finite ring (with unity) and $S$, $T$ be subrings of $R$. Is $S \cup T$ a subring of R? (Counterexamples are easy to find to me when $R$ is an infinite ring or a finite rng.) P.S. I am ...
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2answers
30 views

Inverse Rule for Formal Power Series

I am just really starting to get into formal power series and understanding them. I'm particularly interested in looking at the coefficients generated by the inverse of a formal power series: ...
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0answers
23 views

Classification of algebras over GF(2)

I want to know something about algebras over GF(2). What are best structure theorem that a known at present? What I can read about this? Is there trivial excercise or it is not clear how make it now? ...
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2answers
80 views

On finite commutative rings with the number of ideals equal to the number of elements of the ring

Let $R$ be a finite commutative ring with identity. Under what conditions the number of ideals of $R$ is equal to the number of elements of $R$? The only class of rings with this property that I ...
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2answers
36 views

Is $\mathbb Z[\frac{1+\sqrt 5i}2]$ a ring of fractions of $\mathbb Z[\sqrt 5i]$?

Does there exist a multiplicative set $S\subset \mathbb Z[\sqrt 5i]$ such that $\mathbb Z[\frac{1+\sqrt 5i}2]\cong S^{-1}\mathbb Z[\sqrt 5i]$? Since the multiplicative structure of $\mathbb ...
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1answer
14 views

Is the image of a ring homomorphism an ideal?

Let $\phi : R \rightarrow S$ be a ring homomorphism, then is $im(\phi)$ an ideal in $S$? I ask this because I am studying about modules and in that we say that for a given $R$-module homomorphism the ...
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1answer
106 views

Stable epimorphisms of commutative rings

Recall that an epimorphism $f : A \to B$ in a category with fiber products is called stable (or universal) if for every morphism $C \to B$ the base change $A \times_B C \to C$ is an epimorphism. ...
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9answers
2k views

Are there rings whose multiplicative identity is not the number 1 or number 1-based?

Reading the basic definition of rings, I wondered if there are samples of rings whose multiplicative identity is not the number 1 or number 1-based (for instance the identity matrix is 1-based). ...
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0answers
40 views

Irreducible is prime in the ring $\mathbb{Z} + X\, \mathbb{Q}[X]$

If a ring is an UFD, its irreducible elements are exactly its prime elements. Show that the reverse is not true. Give a nontrivial counterexample. Hint: Consider the ring $\mathbb{Z} + ...
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21 views

$R = \mathbb{Q}[x,y,z]$ and let bars denote passage to $\mathbb{Q}[x,y,z]/(xy-z^2)$ Prove $\bar{P} = (\bar{x},\bar{z} )$

$R = \mathbb{Q}[x,y,z]$ and let bars denote passage to $\mathbb{Q}[x,y,z]/(xy-z^2)$ Prove $\bar{P} = (\bar{x},\bar{z} )$ is a prime ideal. Show $\bar{xy} \in \bar{P}^{2}$ but that no power of ...
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0answers
28 views

Definition with Euclidean domain

Let $R$ be a Euclidean domain and let $A$ be an ideal of $R.$ Then there exists an element $a_0 \in A$ such that $A$ consists of all $a_0x$ as $x$ ranges over $R.$ I found the above theorem ...