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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3
votes
2answers
134 views

A Bézout UFD is a PID. [duplicate]

Let $R$ be an integral domain and a Noetherian U.F.D. with the following property: for each couple $a,b\in R$ that are not both $0$, and that have no common prime divisor, there are elements $u,v\in ...
0
votes
0answers
42 views

Show that there are finitely many different principal ideals [duplicate]

Let $R$ be a U.F.D. and $0\neq d\in R$. I want to show that there are finitely many different principal ideals that contain the ideal $(d)$. $$$$ We have that $R$ is a U.F.D. iff $\forall r\in ...
0
votes
0answers
10 views

Let $R$ ,$S$ be rings and $\alpha$: $R \to S$ be an isomorphism. Show that if $R$ is a ring with a 1 then so is $S$.

I began using the fact that any element of $S$ can be written as $\alpha(u)$ where $u \in R$. I then considered $a,b \in S$ such that $a = \alpha(1)$ and $b = \alpha(u)$ where $ u\in R$. Then using ...
0
votes
1answer
21 views

Let $F$ be an extension field over $K$ , if $[F(x):K(x)]$ is finite , then is $F$ also a finite extension over $K$ ?

Let $F$ be an extension field over $K$ such that $F(x)$ is a finite extension over $K(x)$ ; then is it true that $[F:K]$ is also finite ? ( I know about the converse , that if $F/K$ is a finite ...
1
vote
1answer
164 views

Reducibility of $x^2+1$ in $\mathbb{Z}_n[x]$ [closed]

I want to prove or disprove the statement: $x^2+1$ is reducible in $\mathbb{Z}_n[x]$ $\iff$ there exists $a$ such that $a^2=-1$ in $\mathbb{Z}_n$. How can I prove or disprove the proposition?
1
vote
0answers
28 views

A simple problem involving $\Bbb Z_3[i] $ and units.

Let $\Bbb Z_3[i] = \{a + bi | a, b ∈ \Bbb Z_3, i^2 = −1\}$ a). How many elements are in $\Bbb Z_3[i]$? Explain. I know that there are 9 elements but I also know that under multiplication this forms ...
0
votes
2answers
47 views

Bézout's Identity of polynomials?

Let $P=X^3−7X+6$, $Q = 2X^2+ 5X − 3$ and $R = X^2 − 9 ∈\mathbb Q[X]$. What are $S$ and $T ∈\mathbb Q[X]$ such that $PS + QT = R$? I have calculate the greatest common divisor of $P,Q,R$ are ...
0
votes
2answers
53 views

Find all the units in the indicated rings.

Find all the units in the indicated rings. $Q[√3]$ : I know that this is somehow related to the set of rational numbers but my book doesn't define the elements of this group and I couldn't find ...
1
vote
3answers
91 views

Showing reducibility of a polynomial in a Discrete Valuation Ring

Let $R$ be a complete discrete valuation ring with uniformiser $\pi$. I would like to show that a polynomial $f$ in $R[X]$ is reducible. Does it suffice to show that $f$ is reducible in ...
2
votes
1answer
40 views

Example of finite field extension where root not separable

My class notes has as theorem (without proof): "Let $K/F$ be finite field extension, with $K=F(\alpha_1,\ldots,\alpha_n)$ and $\alpha_k$ is separable for all $k$. Then $K/F$ is separable". My ...
0
votes
3answers
62 views

Each prime ideal contains an idempotent element

An element of ring $e$ is called idempotent iff $e^2=e$. Let $R$ be a commutative ring that contains the identity element and a non-trivial idempotent element. I want to show that each of ...
0
votes
0answers
26 views

Complex Norms when D = 1 mod 4

Let $D ∈ \mathbb Z$ and let $\alpha ∈ \mathbb C$ be such that $\alpha^2 = D$. Let $\beta = \frac{1+\alpha}{2}$ and $\overline{\beta} = \frac{1-\alpha}{2}$ if $D = 1$ mod $4$ and $\beta = \alpha$, ...
1
vote
1answer
50 views

There is no field with exactly 6 elements

I saw the related posts, and I tried a different proof. Please have a look. Let $D$ be any field with $|D|=6$. $|D|=6<\infty \Longrightarrow Char(D)\neq 0\Longrightarrow Char(D)=prime\ number$ ...
1
vote
0answers
25 views

Adjoin inverse element to $\mathbb{Z}/12\mathbb{Z}$ [duplicate]

I am looking at describing the ring obtained from $\mathbb{Z}/12\mathbb{Z}$ by adjoining the inverse of 2. I know the final answer is supposed to be that it is isomorphic to $\mathbb{Z}$ but here is ...
0
votes
0answers
35 views

Prove that the following are integral domains. (A question regarding zero divisors).

Prove that $Z[√5] = \{a + b√5 | a, b ∈ Z\}$ is an integral domain. Prove that $Z[√3i] = \{a + b√3i | a, b ∈ Z\}$ is an integral domain. I'm trying to understand how to show that these are true. By ...
0
votes
0answers
15 views

$ \forall a\in U(R) : ord(a)=Char(R) $

Theorem: Let $(R,+,\cdot)$ be a ring with unity $1_R$. Then $$ \forall a\in U(R) : ord(a)=Char(R) $$ Proof: If $ord(a)=n$, $ord(1_R)=m=Char(R)$ then $n1_R=n(a \cdot a^{-1})=(na) \cdot a^{-1}=0_R ...
-3
votes
1answer
43 views

Zero divisors are Nilpotent [closed]

For $p,n \in \mathbb{N}$ with $p$ is a prime then prove that every zero-divisor in $\mathbb{Z}_{p^n}$ is nilpotent.
0
votes
2answers
51 views

Determine whether the indicated set forms a ring: $S = \{A ∈ M(2,\mathbb R)\mid\det A = 0\}$, under matrix addition and multiplication.

Determine whether the indicated set forms a ring under the indicated operations. $S = \{A ∈ M(2,\mathbb R) \mid \det A = 0\}$, under matrix addition and multiplication. I'm not entirely sure ...
0
votes
1answer
24 views

Is the field of fractions of $F[x_1, \dots, x_n]$ a Noetherian Weyl algebra module

Let $F$ be a field of characteristic zero. Let $D_n$ be the Weyl algebra, i.e., $D_n \subset \mathrm{End}_F(F[x_1, \dots, x_n])$ is the submodule generated by $x_i$ and $\partial_i$, $i = 1, \dots, ...
2
votes
1answer
36 views

A non-constant polynomial with odd-integer co-efficients and of even degree , has no rational root?

Let $f(x)$ be a non-constant polynomial in $\mathbb Z[x]$ with odd-integer co-efficients and even degree ; then is it true that $f$ has no rational root ?
1
vote
1answer
19 views

$F$ be a finite field , then are there infinitely many polynomials $f(x) \in F[x]$ such that $f(a)=0 , \forall a \in F$ ?

Let $F$ be a finite field , then is it true that there are infinitely many polynomials $f(x) \in F[x]$ such that $f(a)=0 , \forall a \in F$ ?
0
votes
1answer
39 views

$a$ in ring $R$, adjoin $b$ such that we have relation $b=a$ then will get ring isomorphic to R [duplicate]

I have been told that if we have a ring $R$ and $a$ is an element of $R$ and if we adjoin an element $b$ with the relation $b=a$ then we will get a ring isomorphic to $R$. However, I have not seen a ...
0
votes
3answers
54 views

How to prove these two things commute?

http://www.cims.nyu.edu/~ckent/spring2016/math343/hw9.pdf I've done all of the homework but I'm stuck at the very end. In part 7d (it's really 6d, I don't know why he labeled the pretext to the ...
0
votes
1answer
36 views

Stuck on last part of rings $\mathbb{Z}[x]/(x^2 + 7)$ and $\mathbb{Z}[x]/(2x^2 + 7)$ isomorphic?

I am checking to see if the rings $\mathbb{Z}[x]/(x^2 + 7)$ and $\mathbb{Z}[x]/(2x^2 + 7)$ isomorphic? I want to assume that the two rings are isomorphic and let $f$ be the isomorphism. I can let A = ...
1
vote
1answer
40 views

homomorphism $\mathbb{Z}[x] \rightarrow \mathbb{Z}$ correspondence theorem question

I am looking at the homomorphism $\mathbb{Z}[x] \rightarrow \mathbb{Z}$ that sends $x$ to $1$. I need to explain what the Correspondence Theorem when applied to this map says about the ideals of ...
1
vote
1answer
36 views

Calculation of the unit group of a finite ring

Is there an easy/fast way using GAP to calculate the unit group of a finite ring? For example, the Units command does not work for some finite rings: (I'm using the ...
4
votes
2answers
33 views

Multiplication map for algebras in the sense of Hopf algebra

Let $R$ be a commutative ring with unity and $A$ a ring with unity. In the definition of algebras, we require a multiplication map $\mu: A\otimes A \rightarrow A$ satisfying certain property. Indeed, ...
0
votes
1answer
37 views

Set of nilpotent elements in $\mathbb Z_n$ [duplicate]

An element $r\in R$ is called nilpotent if $r^n=0$ for some integer $n=1,2,\dots $. We have the following: When $r$ is nilpotent then $1-r$ is invertible in $R$. If $R$ is commutative then the ...
1
vote
0answers
36 views

Consider the ring $\mathbb{R}[x]$. Then which of the following quotients is an integral domain?

Which of the following quotient rings is an integral domain: $\mathbb{R}[x]/(x^2+x+1)$ $\mathbb{R}[x]/(x^2+5x+6)$ $\mathbb{R}[x]/(x^3-2)$ $\mathbb{R}[x]/(x^7+1)$ Now we know that $R/(a)$ is an ID ...
-1
votes
1answer
37 views

Each element is invertible [closed]

Let $R$ be a ring and let $I\subseteq R$ the only maximal right ideal of $R$. I want to show that each element $a\in R-I$ is invertible. $I$ is also an ideal. Could you give me some hints ...
0
votes
2answers
69 views

How can we show that $I$ is an ideal? [duplicate]

Let $R$ be a ring and $I$ the set of non-invertible elements of $R$. If $(I,+)$ is an additive subgroup of $(R,+)$, then show that $I$ is an ideal of $R$ and so $R$ is local. $$$$ I have done ...
1
vote
1answer
76 views

Show that $I$ is an ideal

Let $R$ be a ring and $I\subseteq R$ the only maximal right ideal of $R$. I want to show that $I$ is an ideal. To show that $I$ is an ideal, we have to show that $I$ is a left ideal, right? ...
0
votes
0answers
32 views

$(2, 1+\sqrt{-5})$ is a prime ideal in $\mathbb{Z}[\sqrt{-5}]$ [duplicate]

I need a hint (just a hint please, not a full answer) to proving that $(2, 1+\sqrt{-5})$ is a prime ideal in $\mathbb{Z}[\sqrt{-5}]$. I'm trying to prove it via definition of a prime ideal and ...
1
vote
1answer
35 views

Show that $I^{(n)}$ is a primary ideal belonging to $P$ [duplicate]

Let $A$ be a commutative ring with identity, $P$ a proper prime ideal in $A$, $I$ a primary ideal belonging to $P$ and $n$ a positive integer. The ideal $(I^{n})^{ec}$ (extension and contraction being ...
2
votes
0answers
36 views

Principal local Artinian ring is a quotient of discrete valuation ring.

I have seen here the following statement: Let $R$ be a principal local Artinian ring. Clearly the quotient of a discrete valuation ring is such a ring; conversely it is not difficult to show that ...
1
vote
1answer
27 views

Let $A$ be a ring and $m_1,…,m_k$ maximal ideals

Let $A$ be a ring and $m_1,...,m_k$ maximal ideals of $A$, not necessarily different, and $F_i=m_1\cdots m_{i-1}/m_1\cdots m_i$. Because $m_iF_i=0$, $F_i$ can be made into a $A/m_i$-module defining ...
0
votes
1answer
24 views

Prime element definition and example

From mathworld, prime element,b is an element which is nonzero and non unit and if b divides a product in the ring, b divides one of the factors. My question is, $8$ divides $128$ in integer. But, $8$ ...
0
votes
0answers
97 views

$F[x]/(p(x))\cong F$ precisely when degree of $p(x)$ is $1$

Let $F$ be a field, $p(x)\in F[x]$ an irreducible polynomial, and $F[x]/(p(x))$ be the set of equivalence classes modulo $p(x)$. I think that this is true: $F\cong F[x]/(p(x))$ precisely when the ...
2
votes
2answers
43 views

Can we use Eisenstein's Irreducibility Criterion to show that $x^4+1$ is not reducible in Q?

As such: Let $a(x)=x^4+1\in\mathbb{Q}\left[x\right]$. Then choose any prime $p$. By Eisenstein's Criterion, we see that $p\nmid 1$, $p\mid 0$ (since all coefficients of intermediate terms are 0), and ...
0
votes
1answer
51 views

How do I find the ideals in the ring $\mathbb F_3[x]/(x^2+2)$?

Clearly $\{0\}$ and $\mathbb F_3[x]/(x^2+2)$ will be ideals. How would I find the others?
1
vote
1answer
19 views

Multiplicative identity in a monoid ring.

Let $R$ be a ring and $S$ a subset of $R$. I want to prove that $1:S\rightarrow R: s \mapsto 1_R$ is the multiplicative identity in the ring $(R^{(s)},*,+,1,0)$ (with $R^{(S)}$ the subset of $R^S$ ...
0
votes
0answers
43 views

Prove that if R[x] is a PID, then R is a field

I just need someone to check my proof and provide me feedback: Since $R[x]$ is a PID, then the ideal $I = (x-1)$ generated by the polynomial $x-1$ is maximal because it is of degree 1 added to a ...
0
votes
1answer
35 views

Number of zero divisors in a finite nontrivial ring?

Let A and B be finite non trivial rings, show that the ring $A \times B$ contains at least $|A| + |B| - 2$ many zero divisors. multiplication in this case is defined as: $(a,b)\times (c,d) = (a\times ...
0
votes
0answers
28 views

zero divisors in commutative ring with unity?

Suppose A is a commutative ring with unity. prove if a,b ∈ A, with a not equal to 0, prove if a(b + 1 A ) = b(a + a) then a is a zero divisor or b = 1 A. This is what I have so far- just using ...
1
vote
1answer
14 views

If $P/I_kP$ are finitely-generated, is it true that $P/IP$ is finitely-generated where $I=\bigcap I_k$?

I'm looking into some old results on "big projectives'', and trying to understand some steps. Assume that $R$ is a (commutative) ring and $I_1,\ldots,I_n$ are ideals. Let $I=I_1\cap\cdots\cap I_n$ be ...
0
votes
2answers
62 views

Show that if $p(x)$ is reducible in $F[x]$, then $F[x]/(p(x))$ is not an integral domain.

So I know that for it to be an integral domain it has to have the following properties: Commutative Has multiplicative identity No Zero-Divisors and if $p(x)$ is reducible it can be written as ...
1
vote
1answer
32 views

Irreducible factorisation of polynomial over quotient field

Let $F=\mathbb{Z}_3[x]/<x^2+1>$. Factor $x^4+2$ into irreducibles in $F[x]$. I know that $F$ is a field since $x^2+1$ is irreducible. The usual way to find out that a polynomial is irreducible ...
0
votes
1answer
40 views

The Ring extension isomorphic to the field extension

Let $\alpha$ be algebraic over $F$, with $F(\alpha)$ the smallest field containing both $F$ and $\alpha$, and with $F[\alpha]$ the smallest ring containing both $F$ and $\alpha$. I want to show ...
0
votes
0answers
24 views

Center of matrices over a field [duplicate]

I'm trying to find the center of $\mathbb{M}_n(K)$ with $K$ a field. I know what the center would be if $K$ was a ring, but I think this isn't the same for a field $K$. In particular I'm trying to ...
1
vote
1answer
25 views

Is there a non-trivial ordered ring with an “integer-esque” modulo function?

(I'm inspired by this question.) Is there a [not-necessarily-commutative non-simple ordered ring with a 1 that's not equal to 0] which is not isomorphic to the integers but is such that for all ...