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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-1
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0answers
19 views

Find the cardinality of the ring R be the ring $\mathbb{Z[x]}/((x^2+x+1)(x^3+x+1))$ .

Let $R$ be the ring $\mathbb{Z[x]}/((x^2+x+1)(x^3+x+1))$ and $I$ be the ideal generated by $2$ in $R$. What is the cardinality of the ring $R$? Since $I$ is the ideal generated by $2$, so ...
1
vote
2answers
19 views

Ideal generated by an element

Let $R\left [ x \right ]$ denote the set of all polynomials with real coefficients and let A denote the subset of all polynomials with constant term 0. Then A is an ideal of $R\left [ x ...
2
votes
2answers
66 views

Correspondence between nilpotents and between idempotents

It is well-known and easily proved that whenever $R$ is a commutative ring with unity and $S$ is a multiplicative subset of $R$, each ideal of the localization ring $R_S$ is an extended ideal (with ...
5
votes
2answers
102 views

Idempotent ideals in certain commutative rings

Let $R$ be a commutative ring with zero Jacobson radical such that each maximal ideal of $R$ is idempotent. Does it guarantee that each ideal is idempotent? I know only that if each maximal ideal ...
6
votes
1answer
85 views

What other properties follow from having a ring homomorphism to $\mathbb{Z}$?

(All my rings have $1$, and ring homomorphisms preserve $1$.) In $\mathbf{Set},$ the points of an object $X$ can be thought of as arrows from the terminal object $1$ to $X$. So I guess in general, we ...
1
vote
1answer
12 views

Visualising the kernel of a homomorphism and quotient groups

I am trying to represent myself quotient groups and I'm having trouble seeing what the kernel of a homomorphism : $\Phi: G \rightarrow G/H$ is (be it a ring homomorphism or a group homomorphism). I ...
0
votes
0answers
19 views

Let $R$ be a ring, with group of units $U(R)$. If $R$ is a domain, show that $U(R) = U(R[x])$.

Let $R$ be a ring, with group of units $U(R)$. If $R$ is a domain, show that $U(R) = U(R[x])$. Attempt: I know that this is not true in general if $R$ is not a domain (since $R[x]$ has ...
0
votes
1answer
19 views

How many complexes modulo a prime $p$ are of multiplicative order $p^2 - 1$?

If $i = \sqrt{(-1)} \bmod p$, $p$ prime, does not exist, then we can form numbers of the form $a+b i \bmod p$ with multiplicative order $p^2 - 1$. How often do these numbers occur modulo $p$? In ...
2
votes
3answers
50 views

How quickly can we detect if $\sqrt{(-1)}$ exists modulo a prime $p$?

How quickly can we detect if $\sqrt{(-1)}$ exists modulo a prime $p$? In other words, how quickly can we determine if a natural, $n$ exists where $n^2 \equiv -1 \bmod p$? NOTE This $n$ is ...
1
vote
1answer
34 views

Image of ring homomorphism is a subring

Let $\phi : R\to S $ be a ring homomorphism. Definition: $Im\left ( \phi \right )=\left ( R \right )\phi=\left \{ \left ( a \right )\phi \mid a \in R \right \}$ Prove that $Im\left ( ...
2
votes
1answer
305 views

Any ring is integral over the subring of invariants under a finite group action

I need to prove that if $G$ is a finite group that acts on ring $A$, and $A^G$ is the subring consisting of elements of $A$ which are invariant under all $g\in G$, then $A$ is integral over $A^G$. ...
2
votes
1answer
59 views

Prove that $R(+,.)$ is a division ring but I disproved it

QUESTION: Let $R=\left[\begin{matrix}\alpha & \beta \\ \bar\beta & \bar\alpha\end{matrix}\right]\in \mathbf{M_2(\mathbb{C})} $ where $\bar\alpha,\bar\beta$ denote the conjugates ...
-1
votes
1answer
31 views

Nilpotent matrix given nilpotent traces [on hold]

Let R be a conmutative ring and X a two by two matrix. Supose that Tr(X) and Tr(X^2) are nilpotent elements. Prove that 2X is nilpotent. Thanks a lot.
0
votes
0answers
12 views

concept of conjugacy class in a ring

Can we think of a similar concept of a conjugacy class in a ring which satisfies two three properties like conjugacy classes. I think of a set $M_x={xyx^{-1}-y}$ for $x\in R$ and $R$ is a division ...
1
vote
0answers
16 views

Proving that an element of a ring annihilates a module [duplicate]

Let $R$ be a commutative ring with $1$, $M$ be a finitely generated $R$-module, $\mathfrak{i}$ an ideal of $R$, and $\phi$ an $R$-homomorphism such that: $$1.\;\phi(M)\subseteq \mathfrak{i}M=M$$ i. ...
2
votes
2answers
37 views

on finite division subring of a ring

Is there any example of a ring which is not a division ring but any of its subring is a division ring? According to me if $R$ is a ring and $S$ is a division subring then $1\in S$ and hence ...
3
votes
0answers
40 views

There is no irreducible polynomial of largest degree in $\mathbf{F}_q[x]$

I am asked to prove or disprove that given a finite field $\mathbf{F}_q$, the ring $\mathbf{F}_q[x]$ contains irreducible polynomials of arbitrarily large degree. I couldn't think of a reason why this ...
0
votes
1answer
17 views

Splitting field and automorphisms

I know that if $K$ is a field and $f\in K[x]$, then there exists a splitting field of $f$ on $K$. If one has two isomorphic fields $K_1$ and $K_2$ (say $\sigma$ an isomorphism) and $f\in K_1[x]$, ...
2
votes
2answers
47 views

Irreducibility of polynomials in $\mathbf{Z}_p[x]$ - understanding proofs

I am reading through some irreducibility proofs and there's something I don't quite understand: $x^3+2x+1$ is irreducible in $\mathbf{Z}_3[x]:$ no roots in $\mathbf{Z}_3$ and degree $3$ so ...
-1
votes
0answers
33 views

$\mathbb Z[r_1,r_2,…,r_n] =\mathbb Z[\frac 1m]$ [duplicate]

Question Let $r_1,r_2, ...,r_n \in \mathbb Q $. Then $$\mathbb Z[r_1,r_2,...,r_n] =\mathbb Z[\frac 1m]$$ for some integer $m$. I think m must be the least common multiple of the ...
-1
votes
0answers
33 views

Application of generalized Chinese remainder theorem

Question Consider the ring $\mathbb Z[x]$ and define the ideals $I_p=(px-1)$ where p is prime Prove that $\mathbb Z[x]/I_2I_3...I_p$ is isomorphic to $\{\frac{n}{2^{a_2}3^{a_3}...p^{a_p}} | ...
1
vote
0answers
55 views

proving that $8x^3-6x-1$ is irreducible over $\mathbb{Q}$

When considering the impossibility of trisecting the 60 degree angle, one comes across the polynomial $f(x)=8x^3-6x-1$, which I want to prove is irreducible over $\mathbb{Q}$. I reduced the ...
0
votes
0answers
41 views

How many elements are in the field of fractions $\Bbb Z_3(t)$?

As in exercise for my Galois Theory course I am supposed to find the number of elements in the field of fractions $\Bbb Z_3(t)$. I am very confused as to how to approach this question because I ...
1
vote
3answers
50 views

What is $\mathbb{Z_{n}}\left [ x \right ]$

Question: Show that $\mathbb{Z_{n}}\left [ x \right ]$ has characteristic $n$. What does $\mathbb{Z_{n}}\left [ x \right ]$ stands for? I'm very sure this is not the gaussian ring.
3
votes
3answers
47 views

What polynomial maps to $i$ under $\mathbb{Q}[x] \to \mathbb{Q}[x]/(x^2+1) \simeq \mathbb{Q}[i]$?

The rings $\mathbb{Q}[i]$ and $\mathbb{Q}[x]/(x^2+1)$ are isomorphic, and there is a surjective ring homomorphism from $\mathbb{Q}[x]$ to $\mathbb{Q}[x]/(x^2+1)$. Can someone give me an example of ...
-1
votes
2answers
23 views

How to prove the uniqueness of multiplicative identity?

Suppose $i_1, i_2 \in R$ which are multiplicative identity. Let $a$ also be in $R$. Then $a*i_1=a$ which means $a=i_2$. Thus, $i_2*i_1=i_2$. Now $a*i_2=a$, then $a=i_1$ hence $i_1*i_2=i_1$. Now how do ...
2
votes
1answer
29 views

How do I prove the uniqueness of additive identity?

First, suppose $i_1$ and $i_2$ are additive identity in ring R. From the definition of "additive identity" $a+i_1=a$ such that there is $a$ $\in$ $R$, including for $a=i_2$, so $i_2+i_1=i_2$. But ...
5
votes
3answers
4k views

Proving that $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain

We're proving that $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain, using the norm function $$\nu (a + b\sqrt{2} ) = |a^2 - 2b^2|$$ and the first part says that since $\nu (a + b\sqrt{2} ) = |(a + ...
1
vote
2answers
183 views
+100

Show that the sequence is exact

We have that $R$ is a commutative ring. Suppose that $0\rightarrow A\rightarrow B\overset{f}{\rightarrow} C\rightarrow 0$ and $0\rightarrow C\overset{g}\rightarrow D\rightarrow E\rightarrow 0$ are ...
2
votes
1answer
45 views

Is every “prefield” a field?

Definition 0. Call a poset $P$ well-ranked iff it is well-founded, and for all $x \in P$, we have that any two maximal subchains in the lowerset generated by $x$ have the same length. ...
0
votes
0answers
22 views

Has the characterization of non-unique factorizations been studied in a general context?

In this paper, a theory of principalization fields is introduced, that lets the possible factorizations of an element of an algebraic number field be characterized as groupings of the unique ...
-2
votes
1answer
128 views

About two polynomials $f,g$ such that $f=\pm g$

Let $R$ be an infinite commutative ring with unit and with characteristic zero. Assume that $f,g\in R[x_1,...,x_n] $ are nonzero and such that $f(x_1,...,x_n)=s(x_1,...,x_n) g(x_1,...,x_n)$, where ...
2
votes
1answer
36 views

Study of irreducibility for rings that are not integral domains.

The standard definition of an irreducible element is that an element of an integral domain $D$ is irreducible if to can not be written as the factor of two non-unit elements of the ring. However, I ...
0
votes
2answers
129 views

How can we find the prime ideals of $\mathbb{Z}_{12}$? [closed]

I have found that the maximal ideals of the ring $\mathbb{Z}_{12}$ are $(2)$ and $(3)$. Is this correct? How can we find the prime ideals of $\mathbb{Z}_{12}$ ?
4
votes
2answers
306 views

Is normal extension of normal extension always normal?

Let F be a char 0 field, K be a normal extension of F and L be a normal extension of K. Can it be proved or disproved that L is normal extension of F ?
2
votes
1answer
80 views
+50

On an injective ring homomorphism from the ring of continuous functions to the ring of differentiable functions

Let $\phi : C \to D$ be an injective ring homomorphism such that $\phi(1)=1$, where $1$ denotes the constant function $1$ and $C,D$ are the rings of continuous and differentiable functions on ...
1
vote
1answer
35 views

Even functions absorb composition?

If $f(x)$ and $g(x)$ are real functions and $g$ is even, so is $f(g(x))$. Even functions are also closed under addition. I noticed that these are similar properties to those of an ideal of a ring, ...
-3
votes
1answer
56 views

$\mathbb Z[x]/(5x-1)\cong \mathbb Z[1/5]$ [on hold]

Question Prove that the quotient ring $\mathbb Z[x]/(5x-1)$ is isomorphic to the subring $\mathbb Z[\frac{1}{5}]=\{{\frac{n}{5^k}|n \in \mathbb Z, k \in \mathbb Z}\}$ I'm not familiar with ...
1
vote
1answer
83 views

$\mathbb{Z}[x]$ doesn't have principal maximal ideals [on hold]

Prove that $\mathbb{Z}[x]$ doesn't have principal maximal ideals. Please, I need help with this problem. Thanks!
2
votes
2answers
203 views

Integral dependence over rings is transitive

Let $A\subset B\subset C$ be commutative rings. Suppose $B$ is integral over $A$, and $C$ is integral over $B$. Then I want to show that $C$ is integral over $A$. To be integral means that for ...
0
votes
4answers
36 views

What is a proper non-trivial ideal?

Corollary: Let F be a field, Then, F has no proper non-trivial ideals. I apologise for this trivial question. What exactly is a proper non-trivial ideal? Well, non-trivial is defined as not the ...
2
votes
1answer
37 views

Showing no non-zero element is nilpotent in a ring.

Suppose that R is a ring in which $a^{2}=0$ implies that a=0 Show that R has no-non-zero nilpotent element Attempt: Recall that an element x of a ring R is called nilpotent IF there exists some ...
-1
votes
2answers
37 views

Showing ${Z}\left [ d \right ]=\left \{ a+b\sqrt{d}\mid a,b \in \mathbb{Z} \right \}$ is an integral domain

question: Show that $\mathbb{Z}\left [ d \right ]=\left \{ a+b\sqrt{d}\mid a,b \in \mathbb{Z} \right \}$ is an integral domain. By definition, an integral domain is a commutative ring with ...
0
votes
2answers
47 views

Frobenius maps and irreducible functions on finite fields

Let $\mathbb{F}_q$ be a finite field of order $q=p^n$ for some prime $p$ and $n>1$. Suppose both $f(x)=x^2-ax+b$ and $g(x)=x^2-a'x+b'$ are both irreducible. If, assuming that either $a=a'=0$ or ...
1
vote
0answers
29 views

Is $\mathbb{Z}[G^n]$ isomorphic to $\bigoplus_n\mathbb{Z}[G]$?

Let $G$ be a group, $\mathbb{Z}[G]$ be it's group ring and $G^n$ the direct product of $n$ copies of $G$. Is the group ring $\mathbb{Z}[G^n]$ isomorphic to $\bigoplus_n \mathbb{Z}[G]$? If not, is it a ...
2
votes
1answer
18 views

Not free as a bimodule.

Let $R$ be a ring with 1. I am not following why the ring $R$ is free as a right or left module over itself but not as an $R$-bimodule. Clearly for any $r \in R$, $r=1r1$, so one is a basis as a ...
-1
votes
1answer
34 views

Find elements in the Ring [on hold]

Find all invertible elements, all divisors of zero and all nilpotent elements in the ring $R = \left\{ \begin{pmatrix} a & 0\\ b & c \end{pmatrix} \mid \, a, b, c \in \mathbb{R}\right\}$ with ...
1
vote
1answer
48 views

Ring theory. Find the isomorphism

With the help of the theorem of homomorphism for rings, find an isomorphism $\mathbb{Q} [x] / (x^2 - x) \simeq \mathbb{Q} \oplus \mathbb{Q}$, where $\mathbb{Q} \oplus \mathbb{Q} = \{ (q_1, q_2) \mid ...
0
votes
1answer
31 views

Proving that Char of a field is 2.

Let F be a field of order $2^{n}$. Prove that Char(F)=2. I'm stuck with this question after 30 mins and I have to move on. Note that by a certain theorem, this field has characteristic prime. Any ...
0
votes
1answer
30 views