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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4
votes
1answer
37 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, ...
1
vote
2answers
39 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 ...
6
votes
1answer
33 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 ...
1
vote
2answers
27 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 ...
1
vote
0answers
18 views

Noether normalization (example)

How I can find a Noether normalization of the $\mathbb C$-algebra $\dfrac{\mathbb C[x,y,z]}{(xy+z^2,x^2y−xy^3+z^4−1)}$?
0
votes
0answers
25 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 ...
2
votes
0answers
16 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 ...
1
vote
1answer
37 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 $K$?
0
votes
0answers
17 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 ...
1
vote
2answers
43 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 ...
1
vote
1answer
26 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) = ...
2
votes
0answers
50 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 $z$. ...
0
votes
0answers
13 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 ...
1
vote
1answer
17 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 ...
0
votes
1answer
17 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 ...
0
votes
1answer
18 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 $a=0$ ?
1
vote
1answer
34 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 ...
1
vote
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 ...
5
votes
3answers
77 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 ...
3
votes
1answer
21 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: ...
1
vote
0answers
22 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? ...
8
votes
2answers
63 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 ...
2
votes
2answers
31 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 ...
1
vote
1answer
12 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 ...
4
votes
0answers
57 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. ...
19
votes
8answers
1k 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). ...
3
votes
0answers
39 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} + ...
0
votes
0answers
19 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 ...
2
votes
0answers
26 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 ...
1
vote
1answer
72 views
+50

Let $R$ be a commutative ring with $1 \ne 0$, and let $0 \ne e \in R$ be an idempotent element. Prove the following:

Let $R$ be a commutative ring with $1 \ne 0$, and let $0 \ne e \in R$ be an idempotent element. Note that $eR=\{er|r \in R\}$ is also a commutative ring with identity element $e$. (1) If I is an ...
0
votes
1answer
23 views

Let ideal $I$ be generated by polynomial $p(x)$ in the ring $F[x]$ F-field.

Let ideal $I$ be generated by polynomial $p(x)$ in the ring $F[x]$ F-field. Prove: that the polynomials f(x), g(x) are in the same factor class of the ring $\implies f(x)=g(x)(mod\ p(x))$ ...
1
vote
1answer
94 views

Let $R = \mathbb{Z} + x\mathbb{Q}[x]$. Find all the irreducibles in $R$.

Let $R = \mathbb{Z} + x\mathbb{Q}[x] \subset \mathbb{Q}[x]$. Find the irreducibles of $R$. Show that the irreducible elements in $R$ are $\pm p$ for prime integers $p$ and the irreducible ...
2
votes
2answers
40 views

Form of maximal ideals in an algebraicaly closed polynomial ring

I have been trying to prove the following bijection which is a consequence of the nullstellansatz $$\{\text{maximal ideals of }\mathbb{C}[x_1,\dots,x_n] \} \leftrightarrow \{\text{points in ...
0
votes
0answers
33 views

Every polynomial has a root

Let $A$ be a commutatif ring, and $f\in A[T]$ une polynome. Then in the $A$-algebre $B=A[T]/(f)$ the polynomial $f$ has a root, namely $T \mod (f)$, because $f(T)\mod (f)=f(T)\mod (f)=0$. Do you ...
2
votes
0answers
25 views

If in a UFD every maximal ideal is principal then it is a PID

I want to prove that if in a UFD every maximal ideal is principal then it is a PID. My line of attack is: If it is a field i.e. it has no non-zero proper ideal, then we are done. Otherwise ...
0
votes
0answers
17 views

a factorisation of $x^9−x$ into a product of irreducibles in $\mathbf F_3$

Firstly, given the degree of the extension $[\mathbf F_9 : \mathbf F_3]$ is 2. Then, need to write it as a product of irreducible polynomial. What I've got was that $x(x+1)(x-1)(x^2+1)(x^4+1)$ ...
1
vote
2answers
25 views

An example of an ideal of order $12$

Provide an example of an ideal in $R=\mathbb{Z}_6\times\mathbb{Z}_{10}$ that has order $12$, and indicate whether the ideal is a principal ideal (if it is, then identify the generator for the ...
0
votes
1answer
12 views

Let $R$ be an Euclidean Domain, then any two non zero elements $a,b \in R$ have a $\gcd$ and $\gcd(a,b) = ax+by$ for some $x,y \in R$.

Let $R$ be an Euclidean Domain, then any two non zero elements $a,b \in R$ have a $\gcd$ and $\gcd(a,b) = ax+by$ for some $x,y \in R$. I am facing difficulty to do this. Please Help!
1
vote
1answer
38 views

Definition of Local Ring

I was reading some article on local rings, and it gave the following (equivalent) definitions: A ring $R$ is $local$ if it satisfies one of the following equivalent properties (so this needs proving) ...
1
vote
1answer
43 views

Ideals of $\mathbb{Z}_6\times\mathbb{Z}_{10}$

Two things concerning the ring $R=\mathbb{Z}_6\times\mathbb{Z}_{10}$: Identify a subset of $R$ that is a subring of $R$ but not an ideal of $R$. Identify a maximal ideal in $R$. For the first ...
1
vote
2answers
42 views

Rings isomorphic to $\mathbb{Z}_6\times\mathbb{Z}_{10}$

What are five ring properties that hold for each ring that is isomorphic to $R=\mathbb{Z}_6\times\mathbb{Z}_{10}$, but not for every ring? Suppose $Q\approx R$. Then $Q$ has unity, $Q$ is not a ...
3
votes
1answer
91 views
+50

Let $R$ be a Euclidean domain with degree function $\varphi$ and $R$ not a field. If $a \ne 0$ and $b \ne 0$ be two elements in $R$…

Let $R$ be a Euclidean domain with degree function $\varphi$ and $R$ is not a field. Prove the following: (1) Let $a \ne 0$ and $b \ne 0$ be two elements in $R$. Suppose that $a\mid b$ and $b \nmid ...
3
votes
1answer
93 views

Is it true that $R^n\simeq R^m$ as rings implies $m=n$?

Let $R$ be a commutative ring. We know that if $R^n\simeq R^m$ as $R$-modules for some positive integers $n,m$ then $n=m$. But is it still true when they are isomorphic as rings? Thanks!
0
votes
0answers
59 views

Corollary of Gauss's Lemma (polynomials)

I am trying to prove the following result. I have outlined my attempt at a proof but I get stuck. Any help would be welcome! Theorem: Let $R$ be a UFD and let $K$ be its field of fractions. ...
3
votes
4answers
31 views

Proving the existence of a square root of $ -1_{A} $ in a $ 2 $-dimensional unital algebra $ A $ over $ \Bbb{R} $.

Suppose that $ A $ is a $ 2 $-dimensional unital algebra over $ \Bbb{R} $ with a basis $ \{ 1_{A},u \} $, and assume that $ A $ does not have any zero divisors. Show that $ A $ contains an element $ b ...
3
votes
3answers
90 views

Finite commutative ring with more than $\frac{2}{3}$ of its elements idempotent

Suppose that $R$ is a finite commutative ring with identity element, such that more than $\frac{2}{3}$ of elements are idempotent. Prove that all of elements are idempotent. Please give me a ...
2
votes
1answer
26 views

Construction of the discrete valuation ring

Let $K$ be a field. A surjective transformation: $v: K \to \mathbb{Z}\cup\{\infty\}$ is defined as a discrete valuation, if for any $a, b \in K$, the following statements hold true: $v(ab) = v(a) + ...
3
votes
1answer
54 views

Characterize all finite unital rings with only zero divisors

Is it true that for every finite (for simplicity, commutative) ring $R$ in which every element not equal to $1$ is a zero divisor, is isomorphic to the zero ring or $\mathbb{Z}/2\mathbb{Z}$, ...
0
votes
2answers
17 views

What is the number of elements of $\mathbb Z[i] /I $, where $I:=\{a+ib \in \mathbb Z[i] : 2 \mid a-b\}$?

I know that $I:=\{a+ib \in \mathbb Z[i] : 2\mid a-b\}$ is a maximal ideal of $\mathbb Z[i]$. My question is: what is the number of elements of $\mathbb Z[i] /I $? I am totally stuck. Please help ...
2
votes
1answer
73 views

Show that if $x ≡ 1 (\text{mod } λ)$, then $x^3 ≡ 1 (\text{mod } λ^3)$…

Let $λ = (3 + \sqrt{−3})/2 ∈ Q[\sqrt{−3}]$. How do I show that if $x ≡ 1 (\text{mod } λ)$, then $x^3 ≡ 1 (\text{mod } λ^3)$. Also, how do I show that if $x ≡ −1(\text{mod } λ)$, then $x^3 ≡ −1 ...