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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2
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3answers
22 views

$D$ be a UFD , if an element of $D$ is not a square in $D$ then is it true that , that element is not a square in the fraction field of $D$?

Let $D$ be a UFD , , let $F$ be the field of fractions of $D$ , let $a \in D$ be such that $x^2 \ne a , \forall x \in D$ , then is it true that $x^2\ne a ,\forall x \in F$ ? (This problem is ...
0
votes
0answers
12 views

Factoring polynomials when roots are external to the ring

I shall avoid maths script since I'm typing on a mobile, anyway I think I can do without. I have a question about factoring polynomials over a ring. Let's call R the ring in question. It is clear to ...
1
vote
1answer
58 views

Universal property, localization of rings and modules, and initial element in a category

Sorry for the confusing title. I just started learning category theory and am very confused about the concept "universal property". I am not even sure whether my "proof" is a proof or is just a ...
1
vote
1answer
30 views

Delooping of a ring?

I'm no expert on category theory, so the definition of delooping in the nlab article is a bit over my head. However, I do understand the practical idea that we can think of a group $G$ as a one-object ...
1
vote
1answer
36 views

If $R$ is an integral domain with unity having only finitely many subdomains ( not necessarily with unity ) , then is $R$ finite?

If $R$ is an integral domain with unity having only finitely many subdomains ( not necessarily with unity ) , then is it true that $R$ is finite ? ( I know that there are infinite domains with unity ...
-1
votes
2answers
33 views

Polynomial ring, ideals and Spec

Morning everyone, I want some hint about this. i) Determine all ideals of $\frac{\Bbb{R[X]}}{<X^3-1>}$ where $R$ is real set ii)Is $\frac{R[X]}{<X^3-1>}$ integral Domain iii)...
0
votes
1answer
31 views

Rings with exactly two zero-divisors [on hold]

Why rings with only two zero-divisors are $Z_4$ and $Z_2[X]/(X^2)$?
5
votes
1answer
491 views

When Leavitt path algebras are unital

I’ve been taking a seminar course on Leavitt path algebras, and our source material is rather laconic as far as explanations are concerned. I am attempting to justify (to myself) the following claim ...
3
votes
2answers
56 views

Polynomial ring with arbitrarily many variables in ZF

For a given field $k$ and a set $X$ we want to define the ring $k[X]$ of polynomials with $X$ as the set of variables. We do not assume $X$ to be finite. And we want to do this without employing axiom ...
0
votes
2answers
61 views

Let $R$ be a ring with unity. Consider $X^2 - 1$ $\in$ $R[X]$. Then $X^2 - 1$ has at most two roots in R.

Indicate True/False Let $R$ be a ring with unity. Consider $X^2 - 1$ $\in$ $R[X]$. Then $X^2 - 1$ has at most two roots in R. I need a hint to solve this problem. I have tried some common rings ...
0
votes
1answer
55 views

Is it true that $\mathbb{Q}[x,y]/(xy^2-1)\cong\mathbb{Q}(x)[y]/(y^2-\frac 1x)$? [on hold]

I need to show that $(xy^2-1)$ is prime in $\mathbb{Q}[x,y]$ and I tried to consider the isomorphism $$\mathbb{Q}[x,y]/(xy^2-1)\cong\mathbb{Q}(x)[y]/(y^2-\frac 1x).$$ Does it hold? Thank you.
2
votes
3answers
385 views

Show that $M_2(\mathbb{R})$ has no non-trivial two-sided ideals

In addition to the title question, I also want to find a non-trivial right ideal and a non-trivial left ideal of $M_2(\mathbb{R})$ . Attempt of title question: Suppose $\exists I\subset M_2(\mathbb{...
2
votes
2answers
296 views

Prime implies irreducible

In a unique factorization ring with unity (I am not considering commutativity and zero divisors in definition of UFD) irreducible implies prime. And it was proved in ring with unity without zero ...
3
votes
1answer
143 views

Irreducible and prime elements

In my commutative algebra lecture notes it says: A non-zero element $p$ of a ring $R$ which is not a unit of $R$ is called a prime element if $p=ab$ implies $a$ is a unit or $b$ is a unit. Is this ...
11
votes
2answers
378 views

Are integers mod n a unique factorization domain?

I am trying to learn abstract algebra from scratch, jolly stuff, but in the process of doing so this puzzles me: Having a ring of integers mod $n$, where $n=pq$ is composite, as I understand we have ...
6
votes
0answers
142 views

Can we characterize all infinite Euclidean-domains having exactly one invertible element?

$\mathbb Z_2$ and $\mathbb Z_2[x]$ are two euclidean-domains having exactly one invertible element ; my question is ; Can we characterize all euclidean domains $D$ having exactly one invertible ...
0
votes
0answers
18 views

An example of a module that have no supplement.

We see that if R/J is not coclosed coprojective and J has a supplement then R/J is projective. Now we are looking for an example J has no supplement and also R/J is not coclosed coprojective. But we ...
0
votes
1answer
22 views

Showing that $\mathrm{Rad}((0)) ≠ (0)$ implies $R^\times \subsetneq R[X]^\times$

Let $R$ be a commutative ring with $1$, and let $I ≤ R$ be an ideal. We call $\mathrm{Rad}(I) := \{r \in R: \exists n \in \mathbb{N}_0: r^n \in I\}$ the radical of $I$. I now want to show that if $\...
1
vote
2answers
75 views

Show that $R$ is a field

Let $R$ be a commutative ring with unit. If $R\neq 0$ such that each finitely generated $R$-module is free then $R$ is a field. In my notes there is the following proof: We need to show that ...
12
votes
1answer
106 views

Does there exist a polynomial $p(x) \in \mathbb C[x]$ such that $p(x) \notin \mathbb R[x]$ and $p(x)p(-x)=p(x^2)$?

Does there exist a polynomial $p(x) \in \mathbb C[x]$ such that $p(x) \notin \mathbb R[x]$ and $p(x)p(-x)=p(x^2)$ ? I have noticed that if $a_n$ is the leading co-efficient of $p(x)$ then $a_n=(-1)^n ...
1
vote
2answers
115 views

Specific basis of A-algebra B that is also a free A-module of finite rank.

I have a problem that seems (at least to me) harder then I initially thought. Let $B$ be an $A$-algebra that is also a free $A$-module of finite rank (if necessary we can assume that $B$ is ...
1
vote
1answer
41 views

$k[x,y,z]/(y-x^2,z-x^3)\cong k[x]$, where $k$ is a field

This is generalizing from a previous question, which asks to prove that $k[x,y]/(y-x^2)\cong k[x]$. The way I proved that was by using the homomorphism $\phi:k[x,y]/(y-x^2)\to k[x]$, $\phi(\overline{f(...
2
votes
1answer
51 views

Do there exist semi-local Noetherian rings with infinite Krull dimension?

Do there exist semi-local Noetherian rings with infinite Krull dimension? As far as I know, Nagata's counterexample to the finite dimensionality for general Noetherian rings is not semi-local.
10
votes
2answers
505 views

Existence of prime ideals in rings without identity

Let $R$ be a commutative ring (not necessarily containing $1$). Say that $R$ is the trivial ring if it has trivial (zero) multiplication. If $R$ is the trivial ring, then $R$ has no prime ideals (as ...
3
votes
2answers
145 views

An example of a ring without identity that does not contain any maximal ideal. [duplicate]

I'm trying to find an example of a ring without identity that does not contain any maximal ideal. Help me some hints.
2
votes
1answer
137 views

$R$ be a ring without identity. If $R$ has a maximal left ideal, then the Jacobson radical is still the intersection of all the maximal left ideal?

We know that the definition of the Jacobson radical $J(R)$ (a) in a ring $R$ with identity is: $$J(R)=\cap \mbox{ maximal left ideals}.$$ (b) in a ring $R$ without identity is: $$J(R)=\{a\in R\mid ...
4
votes
1answer
113 views

Example of a finite ring with identity containing a ring without identity

What is an example of a finite ring $R$ with unity and a subring $S$ of $R$ that is not a ring with unity?
4
votes
2answers
139 views

Nontrivial subring with identity of a ring without identity [duplicate]

I'm looking for an example a ring and a subring with $R \subset S$ such that $R$ has 1 but $S$ does not. Its easy to choose R to be the trivial ring with $0=1$, but are there any more exotic examples ...
3
votes
1answer
100 views

In a commutative ring without identity, is $(a)(b)\subset (ab)$ or $(ab)\subset (a)(b)$?

Let $R$ be a commutative ring without unity. Consider an ideal $(a)$ generated by $a\in R$. Note that $(a)=\{ra+na : r\in R, n\in \textbf Z\}$ since $R$ has no identity. I wonder if $(a)(b)\subset (ab)...
8
votes
5answers
1k views

Commutative rings without assuming identity

I was going through Exercises in Dummit&Foote, which does not assume identity in the definition of a ring, and reached the following exercise: Prove that in a Boolean ring ($a^2 = a$ for all $...
6
votes
2answers
119 views

Books on Rings without Identity

I was just wondering if anybody knows of any good books or articles that study rings (and algebras) without (or not necessarily with) identity. I have gone through Thomas Hungerford's Algebra textbook ...
0
votes
2answers
88 views

Is the ring $m\mathbb{Z}$ isomorphic to the ring $n\mathbb{Z}$?

I came over a question in ring theory which I am not being able to proceed upon: When is the ring $m\mathbb{Z}$ isomorphic to the ring $n\mathbb{Z}$, where $m, n \in \mathbb{N}$? I know that to ...
2
votes
1answer
48 views

Are there different left and right ideals in a ring without identity?

For a non commutative ring without identity, is it possible that there will be right and left ideals which are different?
0
votes
1answer
24 views

In an $\Bbb{N}$-graded domain $A$, units are homogeneous

Let $A$ be a graded domain, with additive subgroups $A_n,\,\forall\,n\geq 0$, s.t. ${A_n\cdot A_m}\subseteq A_{n+m}\,\forall\,n,m\geq 0$, and $A=\bigoplus_{n=0}^\infty\, A_n$ as abelian groups. I wish ...
0
votes
0answers
28 views

Describe the prime elements of the ring $\mathbb Z[\sqrt{-2}]$ [duplicate]

I have a ring $\mathbb Z[\sqrt{-2}]$ and I need to describe all the prime numbers of that ring. How I can do that? Thank you
0
votes
2answers
81 views

Polynomial with infinite roots

I started Ring theory recently and I came across this statement while reading polynomial rings.. If $F$ is an infinite field and let $f(x)\in F[x]$ . If $f(a)=0$ for infinitely many elements $a$ ...
0
votes
1answer
43 views

If for all $a\in A$, $a\neq 0$ exists $b \in A$ such that $ab\neq 0$ then prove that $A\cong T_A$.

$A$ is a commutative ring with identity, and $T_A =\{ T_a \mid a\in A\}$, where $T_a=ax$ for all $x\in A$. If for all $a\in A$, $a\neq 0$ exists $b \in A$ such that $ab\neq 0$ then prove that $A\...
3
votes
6answers
1k views

$a^m=b^m$ and $a^n=b^n$ imply $a=b$

Let $D$ be an integral domain and let $a^m=b^m$ and $a^n=b^n$ where $m$ and $n$ are relatively prime integers, $a,b \in D$. How do I show $a=b$?
8
votes
4answers
1k views

Prove that $a=b$, where $a$ and $b$ are elements of the integral domain $D$ [duplicate]

Let $D$ be an integral domain and $a,~b \in D$. Suppose that $a^n=b^n$ and $a^m=b^m$ for any two some $m,~n$ such that $(m,n)=1$. Prove that $a=b$. I know that $ab≠0$ since $D$ contains no ...
0
votes
1answer
52 views

Each automorphism is of that form

Let $R$ be a commutative ring and $c,b\in R$ with $c$ invertible. The correspondence $x\rightarrow cx+b$ defines an unique automophism of $R[x]$ that is the identity in $R$. If $D$ is an integral ...
1
vote
2answers
208 views

About second uniqueness primary decomposition theorem

I'm self-learning commutative algebra from Introduction to Commutative Algebra of Atiyah and Macdonald and get frustrated about the second uniqueness primary decomposition theorem. I copy the theorem ...
3
votes
2answers
88 views

Number of ring homomorphisms $\mathbb{Z}/a\mathbb{Z} \times \mathbb{Z}/a \mathbb{Z} \to \mathbb{Z}/b\mathbb{Z}$, $a,b \in \mathbb{P}$

Number of ring homomorphisms $\mathbb{Z}/a\mathbb{Z} \times \mathbb{Z}/a \mathbb{Z} \to \mathbb{Z}/b\mathbb{Z}$, $a,b \in \mathbb{P}$. I thought about using the fact, that the kernel of a ring ...
2
votes
1answer
36 views

If $a$ and $b$ are elements in a ring with $a^n=b^n$ and $a^m=b^m$ then $a=b$

I was doing the first exercises from the book Exercises in Basic Ring Theory by G. Călugărescu and P. Hamburg and I found one whose solution isn't quite clear to me. Ex. 1.4 If $a$, $b$ are ...
1
vote
1answer
45 views

Exercise on the ring $\mathbb Z \times \mathbb Z$ and its quotient with an ideal

Let $A = \mathbb Z \times \mathbb Z$ a ring, where operations are defined elementwise. a) Prove that the ideal $I$ generated by $x = (4,6)$ is not maximal. b) Find in $A$ (if it exists) an ...
7
votes
0answers
42 views

Maximal ideals in the ring of measurable functions

The $R$ ring of continuous functions from $[0,1]$ to $\mathbb{R}$ has a property that its maximal look like a subset of $R$ consisting of those functions which vanish at a common single point in $[0,...
-1
votes
0answers
31 views

Principal Ideal in a polynomial ring [duplicate]

Let $K[x]$ be a polynomial ring. If I am given two polynomials $P_1$ and $P_2$, and if I find the generator of the ideal of those two polynomials, How can I tell whether or not that ideal is principal?...
2
votes
1answer
23 views

Given a ring $R$ and a ring extension $R'$, if $r=r's$ where $r'\in R'\setminus R$, does that mean that $s\not\mid r$ in $R$?

Given a ring $R$ and a ring extension $R'$, if $r=r's$ where $r'\in R'\setminus R$ and $r,s\in R\setminus\{0\}$, does that mean that $s\not\mid r$ in $R$? I was thinking for example in $\Bbb{Z}$, ...
1
vote
1answer
100 views

Prove that $\Bbb{R}[\cos(\theta),\sin(\theta)]\cong\Bbb{R}[x,y]/(1-x^2-y^2)$ [duplicate]

More precisely, given the ring homomorphism $\phi:\Bbb{R}[x,y]\to\Bbb{R}^\Bbb{R}$, with $\phi(f(x,y)):\Bbb{R}\to\Bbb{R},\,\,\phi(f(x,y))(\theta)=f(\cos(\theta),\sin(\theta))$, where $\Bbb{R}[x,y]$ is ...
1
vote
5answers
309 views

Maximal ideal in the ring of polynomials over $\mathbb Z$

Let $\mathbb Z[x]$ the ring of polynomials with integers coefficients in one variable and $I =\langle 5,x^2 + 2\rangle$, how can I prove that $I$ is maximal ideal. I tried first see that $5$ and $x^...