This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

learn more… | top users | synonyms (1)

-2
votes
0answers
21 views

a question about abstract algebra,prove that the ring is commutative.

(1)A ring R is a booleean ring if for every $a\in R$,$a^2=a$. Show that every Boolean ring is a commutative ring. (2)Let R be a ring,where $a^3=a$ for all $a\in R$.Prove that R must be a commutative ...
0
votes
0answers
17 views

>Let $R$ be an Euclidean ring. Prove that $\operatorname{lcm}(a,b) = \dfrac{ab}{\gcd(a,b)}$ for $a,b \in R.$

Let $R$ be an Euclidean ring. Prove that $\operatorname{lcm}(a,b) = \dfrac{ab}{\gcd(a,b)}$ for $a,b \in R.$ Does this proof work? Any criticism appreciated: We rewrite $\operatorname{lcm}(a,b) = ...
6
votes
0answers
18 views

The difference between the ring version and module version of Chinese Remainder Thereom.

Chinese Remainder Theorem for Commutative Rings If $R$ is a commutative ring with $1$ and $I, J$ are ideals of $R$ that are pairwise coprime or comaximal (meaning $I + J = R$), then $IJ = I \cap J$, ...
0
votes
0answers
7 views

Criterions for $U_1(\mathbb{Z}G)=G$ i.e. units to be trivial in $\mathbb{Z}G$

Notation- $U_1(\mathbb{Z}G)$ here means units of $\mathbb{Z}G$ with augmentation $1$, i.e. $u=\sum_{g\in G} a(g)g \in U(\mathbb{Z}G)$ such that $\sum_{g \in G} a(g)=1$ 1) I have done theorem by ...
2
votes
1answer
46 views

Is the zero ideal $\{0_{M_2(\mathbb{R})}\}$ maximal in $M_{2}(\mathbb{R})$?

Consider the ring $M_2(\mathbb{R})$ of all 2 × 2 real matrices over $\mathbb{R}$. Is the zero ideal $\{0_{M_2(\mathbb{R})}\}$ maximal in $M_{2}(\mathbb{R})$? Is the quotient ring ...
2
votes
2answers
39 views

R is commutative ring with identity & define $\circ$ on $R$ by for any $a,b \in R$ $a \circ b=a+b-ab$ Prove the following

Let R be a commutative ring with identity. Define a new operation $\circ$ on $R$ by for any $a,b \in R$ $$a \circ b=a+b-ab$$ a) Prove that $\circ$ is associative b) Prove that R is a field iff the ...
0
votes
1answer
16 views

Show that $\bar{a}_{n}(\bar{x})^n+···+\bar{a}_{1}\bar{x}+\bar{a}_{0}=0_{F[x]/I}$

Let $F$ be a field, $f(x)$ be an irreducible polynomial in $F[x]$ and $I =(f(x))$. Let $f(x)= a_nx^n+···+a_1x+a_0, a_i \in F$ for $i=0,...,n$. And, $\bar{x} = x + I ∈ F[x]/I$ and $\bar{a_i} = a_i + I ...
6
votes
2answers
77 views

Let R be a ring. Prove that if $x^2=x$ for each $x \in R$, then R is a commutative ring. [duplicate]

Let R be a ring. Prove that if $x^2=x$ for each $x \in R$, then R is a commutative ring. Ok, so I'm just looking for some confirmation that I'm doing this correctly. If we suppose $x,y \in R$ Let's ...
2
votes
1answer
45 views

Algebraic and transcendental functions over $\mathbb Z_n$ — is this a known result?

I have proved a result that seems (to me) interesting, and I am wondering whether it is a known result, and if not whether it seems interesting to others. The result is as follows: Let $R=\mathbb Z ...
5
votes
1answer
61 views

If $R/I \times R/J$ is isomorphic to $R/(I\cap J)$ as $ R $-modules, then $I + J = R$. [duplicate]

If $R$ is a commutative ring with identity and $I$ and $J$ are ideals of $R$ such that $R/I \times R/J$ is isomorphic to $R/(I\cap J)$ as $R$-modules, then $I + J = R$. I know this is the ...
4
votes
3answers
54 views

Example of a ring where all but two of its elements are units

One way of viewing a field is just as a ring where all but one of its elements (namely $0$) is a unit. I'm looking for rings (commutative with a 1) where all but two of its elements are units. I found ...
0
votes
4answers
60 views

Characteristic of a Finite Integral Domain

I am a little confused as how to approach this problem. The title of this problem is the title of the section which it comes from. However, there is no information that the given integral domain is ...
1
vote
0answers
33 views

A quick way to determine number of ring homomorphism

Find the number of ring homomorphism from $\mathbb Z_2\times \mathbb Z_2$ to $\mathbb Z_4$. I know that checking idempotent elements the number of ring homomorphisms is $1$. Again the number of ring ...
3
votes
2answers
41 views

If $x = a +bi$ is not a unit of $\mathbb{Z}[i],$ prove that $a^2+b^2>1.$

A problem from my algebra text: If $x = a +bi$ is not a unit of $\mathbb{Z}[i],$ prove that $a^2+b^2>1.$ I think it's false since $x = 0 + 0i = 0 \in \mathbb{Z}[i]$ is not a unit, but $0 + 0 ...
0
votes
1answer
22 views

A doubt in a lemma on integral group rings.

In a paper by Farkas, I was doing this lemma, where I had this doubt (red underlined) in the proof of the lemma. Can anybody explain me how does it follow $\alpha$ is centralized by $H$. It should ...
0
votes
0answers
26 views

Quotient of Ideals in matrix rings

I'd like to know where could I find some info about the quotient $I:J=\{a\in R\mid aJ\subseteq I\}$ ($R$ a ring) in matrix rings? Or for example, in a matrix ring over $\mathbb{Z}$. I would like to ...
2
votes
1answer
25 views

How to classify the rings of fractions of a principal ideal domain?

Let $A$ be a principal ideal domain and let $K$ be its field of fractions. I proved a) Every ring $B$ such that $A \subset B \subset K$ is a ring of fractions of $A$, and c) Show that any ring of ...
1
vote
1answer
34 views

Prove that $\Bbb{Z}[i]/I$ is finite where I is an ideal of $\Bbb{Z}[i]$

Show that for any nontrivial ideal $I$ of $\Bbb{Z}[i]$, $\Bbb{Z}[i]/I$ is finite. $\Bbb{Z}[i]$ is a PID, so $I=\langle{a+ib\rangle}$. Now $\Bbb{Z}[i]/I$ has elements of the form ...
0
votes
1answer
24 views

Modules over Itself

Let $F$ be a field and let $R$ be the ring of polynomials over $F$ in infinitely many variables $x_1, x_2 , \cdots x_n \cdots$ (Of course, each element of $F$, being a polynomial, will involve only ...
1
vote
1answer
29 views

When is $\mathbb{Z}[\sqrt{d}]$ an Euclidean domain?

Where $d \in \mathbb{Z}$ is not a perfect square. This problem appeared in our exam and now I'm asking how was I supposed to answer?
0
votes
0answers
50 views

Prove $\mathbb{Z} [\sqrt{2}]$ is a Euclidean ring.

The definition of Euclidean ring: An integral domain R is called Euclidean ring if $\exists \delta$ : $R${$0$} -> $\mathbb{N} \cup{0}$ satisfying: (1) $\delta (a) \leqslant \delta (ab)$ if a, b $\in ...
0
votes
1answer
15 views

$A$ prime in $S$ implies that $\phi^{-1}(A)$ prime in $R$ ; $A$ maximal in $S$ implies that $\phi^{-1}(A)$ maximal in $R$

Suppose $R,S$ are commutative rings with unities. Let $\phi$ be a ring homomorphism mapping $R\to S$ and let $A\subset S$ be an ideal. How can I start the proofs for: Showing that $A$ prime in ...
1
vote
2answers
25 views

If $R$ is a ring with unit element $1$ and $\varphi$ is a homomorphism of $R$ onto $R'$, prove that $\varphi(1)$ is the unit element of $R'$.

Herstein 3.4.20: If $R$ is a ring with unit element $1$ and $\varphi$ is a homomorphism of $R$ onto $R'$, prove that $\varphi(1)$ is the unit element of $R'$. I don't understand why $\varphi$ needs ...
0
votes
2answers
53 views

Show that $Im(\phi) = \mathbb{Z}[i]$

Let $\phi: \mathbb{Z}[x]\to \mathbb{C}$ and $\phi(f(x)) = f(i), \forall f(x) \in \mathbb{Z}[x].$ Show that $Im(\phi) = \mathbb{Z}[i]$ My attempt: I am not sure if it's correct: First, we need to ...
0
votes
0answers
19 views

Jacobson radical of polynomial ring [duplicate]

Let $R$ be a ring, i want to show that: if R has not nil-ideals than $J(R[x]) = \left\{ \emptyset \right\}, \text{where $J$ Jacobson radical, $R[x]$ - polynomial ring over $R$}$
3
votes
3answers
65 views

Is $ \langle x,5 \rangle $ a maximal ideal of $ \mathbb{Z}[x] $?

Here, $ \langle x,5 \rangle $ is the ideal generated by $ x $ and $ 5 $ in $ \mathbb{Z}[x] $, which is the polynomial ring over $ \mathbb{Z} $. How should I approach this question?
3
votes
1answer
46 views

Let $a$, $b$, and $c$ be elements of a commutative ring, and suppose $a$ is a unit. Prove that $b$ divides $c$ if and only if $ab$ divides $c$.

Let $a$, $b$, and $c$ be elements of a commutative ring $R$, and suppose $a$ is a unit. Prove that $b$ divides $c$ if and only if $ab$ divides $c$. Okay so here is the proof I came up with. Please be ...
0
votes
2answers
55 views

Let $R$ be a finite ring with unity. Prove that $x$ is a LZD $\iff$ x is a RZD

Let $R$ be a finite ring with unity. Let $x \in R$. Prove that $x$ is a Left Zero Divisor $\iff$ x is a Right Zero Divisor. My attempt Suppose $x$ is a LZD. Then, $\exists y \in R$ such that $xy = ...
1
vote
0answers
45 views

A ring with a left cancellable element and a right identity always has an identity.

Let $R$ be a ring with $a, e \in R$ such that $a$ is not a left zero-divisor and $be=b, \forall b \in R$. Prove that $R$ has an identity. My attempt Let, $aeb = ab \Rightarrow aeb - ab = 0 ...
1
vote
0answers
43 views

Is the ring of germs of $C^\infty$ functions at $0$ Noetherian?

I'm considering the property of the ring $R:=C^\infty(\mathbb R)/I$, where $I$ is the ideal of all smooth functions that vanish at a neighborhood of $0$. I find that $R$ is a local ring of which the ...
2
votes
1answer
32 views

To prove that $I = \{\,(n,m) \in \Bbb Z \times \Bbb Z : n,m $ are even $\}$ is not a maximal ideal of $ \Bbb Z \times \Bbb Z $.

To prove that $I = \{\,(n,m) \in \Bbb Z \times \Bbb Z : n,m $ are even $\}$ is not a maximal ideal of $ \Bbb Z \times \Bbb Z $. Thus there exists an ideal $J$ of $ \Bbb Z \times \Bbb Z $ such that $I ...
2
votes
1answer
31 views

Ring Homomorphism from $\mathbb{Z}_m$ to $\mathbb{Z}_n$

Suppose $R$ is a ring homomorphism from $\Bbb{Z}_m$ to $\Bbb{Z}_n$ , prove that if $R(1) = a$ then $(a^2)=a$. Also show, its converse is not true. The first part goes like this : $R(1) = a , ...
2
votes
1answer
37 views

Proving every element of $\mathbb{F}[x]/(p(x))$ can be expressed uniquely

Proving every element of $\mathbb{F}[x]/(p(x))$ can be expressed uniquely in the form $a(x) + (p(x))$ where $\text{deg}(a) < \text{deg}(p)$ this is a homework problem and I'm stuck, here is my ...
0
votes
2answers
34 views

Let $I = \{a +\sqrt2b \in \Bbb Z[\sqrt2] : a$ and $b$ are both multiple of $5\}$. [duplicate]

Let $I = \{a +\sqrt2b \in \Bbb Z[\sqrt2] : a$ and $b$ are both multiple of $5\}$. Show that $I$ is a ideal.(I have done it) But how to show that it is maximal?
1
vote
3answers
33 views

Elements of $\mathbb{Z}/(n)$

Let $(n) = \{ \lambda n | \lambda \in \mathbb{Z} \}$. In my book it has shown that every element in $\mathbb{Z}/(n)$ can be expressed uniquely in the form $r + (n)$ where $0 \leq r \leq n-1$ now I ...
1
vote
1answer
13 views

Commutative rings and ideals, showing a map is well defined

Let $R$ be a commutative ring with an ideal $I$. The additive group $R/I$ is the set of cosets of $I$ with respect to addition in $R$. Let $\cdot : R/I \times R/I \to R/I$ be defined by ...
2
votes
1answer
20 views

Proving that a Left semisimple ring $R$ is both left noetherian and left artinian

Prove that a left semi-simple ring $R$ is both left noetherian and left-artinian. I am following the proof given in pg 27,A first course in non-commutative rings (T.Y.Lam). Its strategy is to show ...
0
votes
1answer
46 views

Let $I = \{a +ib \in \Bbb Z[i] : a$ and $b$ are both multiple of $5\}$.

Let $I = \{a +ib \in \Bbb Z[i] : a$ and $b$ are both multiple of $5\}$. Show that $I$ is an ideal of $\Bbb Z[i]$. Is $I$ a maximal ideal? And to find the numbers of elements of the quotient ring $\Bbb ...
1
vote
1answer
45 views

An approach to proving that $\Bbb{Q}[x,y]/(x^3-y^2)$ is isomorphic to $\Bbb{Q}[t^2,t^3]$

I have to prove that $\Bbb{Q}[x,y]/(x^3-y^2)$ is isomorphic to $\Bbb{Q}[t^2,t^3]$. My approach: Let us consider $t^2$ and $t^3$ as separate variables $x$ and $y$. The relations that hold for them ...
1
vote
1answer
24 views

Unique factorization consequence

Let R be principal ideal domain and $p$ prime element and $b \in R$, $b\neq 0$, $E=R/(pb)$ module over $R$ and let $E_p$ be submodule consisting of elements with exponent $p$ (there exist positive ...
3
votes
1answer
30 views

If C is a chain of non-principal ideals and the union of the ideals of the chain contains a generator, why is the union then principal

If C is a chain of non-principal ideals and the union of the ideals of the chain contains a generator why is the union then principal. I understand this is a contradiction already and seems painfully ...
0
votes
0answers
23 views

direct sum of modules is isomorphic to the direct sum permuting indices? [on hold]

the primary for the exercise idea is to use the universal property of the external direct sum of modules.
0
votes
0answers
28 views

Verification: Let $R$ be a unique factorization domain and let $a,b \in R$ be nonzero, non-unit elements. Prove that $a$ and $b$ have a LCM

Let $R$ be a unique factorization domain and let $a,b \in R$ be nonzero, non-unit elements. Prove that $a$ and $b$ have a least common multiple, and describe one such multiple in terms of the ...
0
votes
2answers
24 views

The mapping $\theta : S^{-1}R \rightarrow (\pi(S))^{-1}(R/I)$ is a well-defined ring epimorphism.

I'm working on this problem for a homework assignment. Note that $R$ is a commutative ring with unity, $I$ is an ideal of $R$, and $\pi : R \to R/I$ is the canonical projection given by $\pi(r)=r+I$. ...
0
votes
2answers
20 views

Show that if $F$ is a field, then $<x>$ is maximal in $F[x]$. Also, show that $F[x]$ is not local.

See statement above. So far I have the following: Assume that $<x>$ is not maximal. Then $ <x> \subset <f(x)> \neq F[x]$. This means that $x = f(x) g(x)$. Since $x$ is ...
1
vote
2answers
39 views

Prove that $\mathbb{Z}_{mn}$ has atleast four idempotent elements. [duplicate]

Suppose $m, n > 1$ are positive integers which are relatively prime. Prove that $\mathbb{Z}_{mn}$ has atleast four idempotent elements. Two of them are $[0], [1]$, how will I find the other two?
3
votes
0answers
91 views

Cardinomials: Like cardinalities, but polynomial valued

I want to see if this notion is known (or if it makes sense). Let $F$ be a field. Let $A$ be a finite dimensional commutative unital algebra over $F$. Let $X_1$, $X_2 \in A$ etc. be such that their ...
0
votes
2answers
33 views

Is it true that the order of any quotient ring $\mathbb Z[i]/\langle a+ib \rangle $ is $a^2+b^2$ ? (where not both $a,b$ are zero)

Is it true that the order of any quotient ring $\mathbb Z[i]/\langle a+ib \rangle $ is $a^2+b^2$ ? ( I know it is atmost finite ) Please help . Thanks in advance .
1
vote
1answer
26 views

On non-constant multiplicative norms on integral domain and when does the absolute value of the norm is unity implies the element is unit?

Consider $\mathbb Z[\sqrt {d}]$, where $d$ is any non - square integer, define $$N(a + \sqrt d b) = a^2 - db^2 = (a + \sqrt d b)(a - \sqrt d b)$$ as $\mathbb Z \subseteq \mathbb Z[\sqrt {d}]$, so from ...
1
vote
1answer
28 views

Proof of that in an integral domain, every prime element is irreducible.

I would like to prove that in an integral domain $R$, every prime element $p$ is irreducible. I understand the case where $p = ab$ but the textbooks I have read do not address the case where $p \neq ...