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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How can we find the lists?

Let $R$ be a commutative ring with unit. We consider the polynomial ring $R=\mathbb{R}[t]$ and the $R$-module $M=\mathbb{R}^3$, where $a\cdot x$ ( $a\in R,x\in M$ ) is defined as usual if $a\in ...
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0answers
19 views

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

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 ...
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3answers
24 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 ...
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1answer
13 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 ...
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0answers
21 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 ...
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1answer
20 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 ...
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3answers
51 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 ...
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1answer
62 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 ...
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0answers
13 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 ...
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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. ...
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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 ...
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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 $f\in K_1[x]$, ...
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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 ...
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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.
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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 ...
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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}} | ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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!
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
2
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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 ...
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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 ...
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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 ...
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1answer
30 views

Prove that a commutative ring without proper ideals is a field [duplicate]

Let $R$ is a commutative ring which has no proper ideals. Prove that $R$ is a field.
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2answers
46 views

Are $\{0\}$ and $\mathbb{Z}$ the only canonically totally-ordered rings?

Definition 0. Given a ring $R$, write $\mathbb{N}_R$ for the set $\{n \cdot 1_R : n \in \mathbb{N}\}.$ So: $$\mathbb{N}_R = \{0_R,1_R,1_R+1_R,\ldots\}$$ Definition 1. Write $\lesssim_R^\mathbb{N}$ ...
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1answer
80 views
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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 ...
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2answers
31 views

The product $J^{n}A$ of ideals

Suppose that $A$ and $J$ are two ideals of a ring $R$. I can't understand the following implication: If $JA = A$ then $J^{n}A = A$ for all $n > 0$. True that $J^{n}A$ is an ideal of $A$ for all ...
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2answers
25 views

Ideals of $\mathbb Z/11\mathbb Z \times \mathbb Z/11\mathbb Z $? [duplicate]

How would one go about finding all of the ideals of this ring? (with addition and multiplication defined component-wise)
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0answers
38 views

Maximum order of element in group of units in a ring

Let $s$ be a natural number and $U(s)$ be a group of units in the ring $\mathbb{Z}/s\mathbb{Z}$. Let $\phi(s) = 2^{k_1}p^{k_2}$, where $p$ is a an odd prime number. I don't understand why the maximum ...
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1answer
29 views

Hilbert function and Hilbert polynomial

I have largely studied Hilbert function and Hilbert polynomial for polynomial rings over fields of characteristic zero. Is it possible to extend the theory also for polynomial rings over fields of ...
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0answers
26 views

Prove the Radical of an Ideal is an Ideal

I am given that $R$ is a commutative ring, $A$ is an ideal of $R$, and $N(A)=\{x\in R\,|\,x^n\in A$ for some $n\}$. I am studying with a group for our comprehensive exam and this problem has us stuck ...
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1answer
43 views

Units of $\mathbb{Z}[\sqrt{7}]$

Let $\mathbb{Z}[\sqrt{7}]=\{a+b\sqrt{7}\mid a,b\in\mathbb{Z}\}$. Let $\mathbb{Z}[\sqrt{7}]$ have the usual addition and multiplication, namely $$(a+b\sqrt{7})+(c+d\sqrt{7})=(a+c)+(b+d)\sqrt{7}$$ and ...
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0answers
14 views

Simple Question about Valuations and Krull Rings

I have what is a very simple question about essential valuations for Krull rings. Before getting to the question, I'll give a sketch of the situation. Any help would be much appreciated. Suppose that ...
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1answer
56 views

Automatic additivity of multiplicative maps

There are results which guarantee that a multiplicative bijection between commutative rings is actually additive (that is, it is a ring isomorphism). For example this result of Martindale initiated ...
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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 ...
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1answer
14 views

Cyclic algebras of degree $4$ and period $2$

Recall that if a field $k$ has a primitive $n$-root of unity $\omega$, then the cyclic $k$-alegbras of degree $n$ (ie of dimension $n^2$) have the following familiar presentation : they are generated ...