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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1answer
25 views

$R$ is a ring with identity. Why from $f(1)=0$ it's concluded that $\forall r\in R; f(r)=0$?

The original question is this: Let $R$ be a ring with identity and $\mathbb{C}$ the ring of complex numbers. Suppose $f,g:R\rightarrow \mathbb{C}$ are two ring homomorphisms such that for every $r$ ...
6
votes
2answers
108 views

The ring $\mathbb{C}[x,y]/\langle xy \rangle$

What can be said about the ring $\mathbb{C}[x,y]/\langle xy \rangle$? I was very certain that $$\mathbb{C}[x,y]/\langle xy \rangle \cong\mathbb C[x] \oplus\mathbb C[y]$$ since the elements in ...
1
vote
0answers
15 views

Finite ring having $2^n-1$ invertible elements [duplicate]

Let $A$ be a ring with $0\neq1$ having $2^n-1$ invertible elements and at most $2^n-1$ non-invertible elements. Then $A$ must be a field? How many elements does the ring need to have? Because $2^n-1$ ...
0
votes
1answer
33 views

Linear endomorphisms of $k(t)$

Let $k$ be a field and let $k(t)$ denote the field of rational variables in $t$. Is it possible to characterize all $k$-linear transformations from $k(t)$ to $k(t)$? Is $End_{k}(k(t)) \cong k(t)$ ?
-6
votes
1answer
101 views

Prove $R$ is a finite ring [on hold]

Suppose that $R$ is a ring and $D(R)$ is the set of all elements $x\in R$ such that there exist $b,c\neq 0$ with $xb=cx=0$. Prove that if $1<|D(R)|<\infty$ then $|R|<\infty$. ($|S|<\infty$ ...
0
votes
2answers
30 views

$M \neq 0$ but $M^* = 0$.

Let $A$ be a ring. For any left, resp. right $A$-module $M$ give the abelian group $\text{Hom}_A(M, A)$ the structure of a ring $A$-module (to be denoted $M^*$), resp. left $A$-module (to be denoted ...
1
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0answers
17 views

Formal power series over a regular ring

I'm trying to prove that if $A$ is a regular ring then so is $A[[X]]$. The only proof i found of this statement is in Commutative Ring Theory by Matsumura, but it seems a bit over my knowledge so I'd ...
2
votes
1answer
45 views

Are all rings $\mathbb{Z}$-modules?

In my course of associative algebra we covered modules and an excercise involved showing that every ring $R$ can also be viewed as an $R$-module. This was straightforward enough. Is it also true that ...
1
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0answers
26 views

Commutative rings of order $p^3$. [duplicate]

Does anyone know of a listing of all rings (or at least those that are commutative) of a given order? In particular $p^3$, for a prime $p$.
0
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0answers
16 views

If $A$ is a noncommutative ring then all biderivation is inner.

Before the question I will post some definitions: Derivation: An additive map $\delta: A \longrightarrow M$, where $A$ is a ring and $M$ is a $(A,A)-$bimodule is called derivation if $\delta(xy) = ...
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0answers
25 views

A Problem for Nil-Ideals

This is actually a homework problem. I need some help. Consider a ring $R$ and $I$ be a finitely generated nil-ideal of $R$. Is $I$ a nilpotent ideal? I have proved this for commutative rings. But for ...
2
votes
0answers
43 views

Ideals in a ring as geometric objects?

I am interested in learing about the possibility of (one-sided) ideals in a ring being repreented geometrically. In other words, about their status as geometric objects (after all, they can be dealt ...
1
vote
1answer
39 views

gcd of $x$ and $2$ in $Z[x]$

In $Z[x]$, $x$ and $2$ has gcd $1$. But they cannot be expressed as the linear combination of two polynomials. Then assuming that $1=2.f(x)+x.g(x)$ we are supposed to arrive at a ...
1
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1answer
26 views

Every ring with $1$ and with no zero divisors and no non-trivial ideals is a division ring

It is well known that every commutative ring with unity $R$ that contains no non-trivial ideal is a field, since given $a \neq 0$, $(a)=R$, therefore there exists $x \in R$ with $ax=xa=1$. What ...
3
votes
1answer
29 views

Non-artinian center

Recall that an Artinian ring is a ring that satisfies the descending chain condition on ideals. What is an example of an Artinian ring whose center is non-Artinian?
1
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1answer
30 views

Jacobson radical of tensor product

Suppose $R$ and $S$ are associative rings with unit and that $J(R)$, the Jacobson radical of $R$, is a nil ideal. Clearly if $R$ is commutative then $J(R)\otimes_\mathbb{Z} S$ is a nil ideal. Is this ...
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0answers
23 views

Composition of module homomorphisms and their kernels

Let $R$ be a principal ideal ring and let $I$ and $J$ be two ideals of $R$. Suppose $\phi: R \times R \rightarrow R \times R$ and $\psi: R \times R \rightarrow R \times R$ are two $R$-module ...
0
votes
0answers
27 views

Kaplansky characterization of principal Artin ring

I would like to learn the proof of this paper of Kaplansky where it is proven that for a commutative ring every module split as sum of cyclic module iff the ring is an Artin principal ideal ring (well ...
3
votes
0answers
31 views

Unit group of an imaginary quadratic ring

Let $R$ be an imaginary quadratic ring. Then, the unit group $R^{\times}$ is finite. To prove this, I worked with normal forms, algebraic integers and the fact that $R \not \subset \mathbb{R}$. But I ...
4
votes
3answers
82 views

For a prime integer $p \in \{2, 3, 5, \cdots\}$, is $pR$ a maximal ideal in $R$?

If $R$ is a commutative ring with unit and $p$ is a prime number ($2,3,5,\cdots$), then is $pR$ a maximal ideal of $R$? If not what conditions should I impose on $R$?
3
votes
1answer
52 views

Prove that $f_n=-1+\prod_{i=1}^{n}(X-i)$ is irreducible in $\mathbb{Z}[X]$

Prove that, for all $n\in \mathbb{N}$, $f_n=-1+\prod_{i=1}^{n}(X-i)$ is irreducible in $\mathbb{Z}[X]$.
0
votes
1answer
46 views
+50

If the Ideals Generated by the Coefficients of $f(X),g(X)$ are $R$, then so is the Ideal Generated by $f(X)g(X)$.

Let $R$ be a commutative ring with unity, and let $f(X),g(X)\in R[X]$. Assume the ideals generated by the coefficients of $f(X),g(X)$ are both $R$. Prove that the ideal generated by the ...
1
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1answer
59 views

If $[A,A]A[\lambda,A] = 0$ then $\lambda \in Z(A).$

Suppose that $A$ is a unital ring and $([A,A]) = A.$ If $[A,A]A[\lambda,A] = 0$ prove that $\lambda \in Z(A).$ Comments: This is part of an exercise I'm doing, I'm posting this part because I am ...
3
votes
0answers
22 views

Module endomorphisms with the same kernel

Let $R$ be a finite commutative principal ideal ring. Let $n$ be a positive integer. For $i=1, \ldots, n-1$ we let \begin{align*} w_i := w_i(x_{i+1}, \ldots, x_n) = \sum_{j=i+1}^n t_{ij}x_j \in ...
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0answers
8 views

Reference for a Dickson Determinant Polynomial

For $2\leq \ell \leq k$, consider the polynomial \begin{equation} P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k] \end{equation} ...
2
votes
2answers
182 views

Divisors of zero in polynomial ring

I have the following theorem: McCoy: Let $R$ be a commutative ring with identity. If $f=\sum_{i=0}^na_iX^i$ is a zero divisor in $R[X]$, then there exists a nonzero $c$ in $R$ such that $cf=0$. ...
0
votes
0answers
30 views

Decomposition and valuation rings

I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theory: $(K,u)$ be a field and its absolute value, $(K_u,\bar u)$ be its completion and absolute value ...
2
votes
1answer
30 views

Modify this formula : $R/I \cong \phi[R]/\phi[I]$

Let $R$ be a ring and $I$ an ideal of $R$, and let $\phi : R\longrightarrow R'$ be a ring homomorphism. Studying by myself, I have a conjecture the following: $$R/I \cong \phi[R]/\phi[I].$$ This ...
2
votes
1answer
25 views

Prove that $R/(2i)$ and $\mathbb{Z}/4\mathbb{Z}$ are isomorphic rings

Let $R=\{a+2ib|a,b\text{ integers}\}$.Prove that $R/(2i)$ is isomorphic to $\mathbb{Z}/4\mathbb{Z}$
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0answers
15 views

Local Constancy of Rank Function

Recently I asked this question. I believe that I have come up with a solution, but I am unsure, because the proof I have seems too easy to be true, and doesn't make very many assumptions. My ...
0
votes
1answer
51 views

Endomorphisms of $\mathbb{Z}/n\mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$

Let G be any abelian group, End(G) be the set of all group homomorphisms $\varphi\colon G\to G $. End(G) is a unital ring under the operations + and $\cdot$(Please refer to the link for detail, ...
0
votes
1answer
37 views

Every prime ideal is maximal [duplicate]

Problem: Show that if R is a finite ring, then every prime ideal of R is maximal. My attempt: Let I be a prime ideal of R. Then, by definition of a prime ideal, ab ∈ I implies a ∈ I or b ∈ I for ...
4
votes
1answer
91 views

finitely generated ideal and number of generators

let $I$ be a finitely generated ideal of ring $R$. Suppose $I/I^2$ as an $R/I$ module is generated by $r$ elements then question is to prove that $I$ is generated by $r+1$ elements.. I have tried ...
2
votes
1answer
21 views

Is there a binary operator (besides composition) closed under permutations or a notion of a metric space on permutations?

When i say "a binary operator closed under permutations" I mean, given $2$ (finite, same number of elements) permutations $p_1$, $p_2$ , is there an operator "$+$" such that $p_1+p_2=p_3$ ($p_3$ a ...
7
votes
1answer
70 views

Is this ring a PID? [closed]

Let $R$ be the $k$-subalgebra of $k(t)$ generated by the set $k[t]$, of all polynomials, and a pair of rational functions: ${1\over{t-1}}$ and ${1\over{t-2}}$. Is the ring $R$ a PID?
1
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1answer
37 views

Identity mapping

Problem: Show that the identity mapping is the only ring homomorphism from $\mathbb{Z}$ to $\mathbb{Z}$. Solution: Let $\varphi:\mathbb{Z}\rightarrow\mathbb{Z}$ be a ring homomorphism. Then ...
1
vote
1answer
52 views

Ring homomorphism

Problem: Let $R$ be a commutative ring, and let $D$ be an integral domain. Let $φ : R → D$ be a nonzero function such that $φ(a+b) = φ(a) + φ(b)$ and $φ(ab) = φ(a)φ(b)$ for all $a,b \in R$. Show ...
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votes
0answers
31 views

Prove that $E(x)y + E(y)x \in Z(A)$ for all $x,y \in A$.

Let $A = M_2(C)$, where $C$ is any commutative ring. Define $E: A \longrightarrow A$ by $E(x) = x - tr(x)Id_2$, where $tr(x)$ denotes the trace of $x$. Prove that $E(x)y + E(y)x \in Z(A)$ for all $x,y ...
0
votes
1answer
38 views

Show that $([A,A])$ contains the identity matrix.

Suppose $A = M_n(C)$, $n \geq 1$, where $C$ is a commutative unital ring. If $n \geq 2$, then $([A,A])$, the ideal generated by $[A,A]$, contains the identity matrix.
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1answer
38 views

characteristic of a ring

I got 4 short question about characteristic. 1) What is characteristic of integral domain D which suffices $20 \cdot 1_D=0_D=12 \cdot 1_D$ 2) Let $A=\{0,1,a\}$ be a integral domain what is ...
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0answers
33 views

Existence of a splitting ring

Let $R$ be a commutative ring and $f\in R[X]$ be a monic non-constant polynomial. How can one show that there exists a commutative ring $S$ so that $R$ is a subring of $S$ and $f$ can be written as a ...
3
votes
1answer
46 views

Adjoining an identity to a ring

I am run into the following in an Algebra text: "Let $R_0=\mathbb Z/2\mathbb Z⊕\mathbb Z/2\mathbb Z⊕\cdots$ viewed as a ring without identity, with addition and multiplication defined componentwise. ...
1
vote
1answer
28 views

Gauss lemma for arbitrary commutative ring [duplicate]

Part (iv) of exercise #2 for chapter 1 in Atiyah and Macdonald's book Introduction to Commutative Algebra asserts that if $f, g \in A[x]$ are primitive then $fg$ is primitive. We know that this is ...
0
votes
1answer
32 views

Minimal polynomial is irreducible

Suppose $\mathbb{E}$ is a field extension of $\mathbb{F}$. If $a$ is algebraic over $\mathbb{F}$ we define the minimal polynomial for $a$ as the monic irreducible generator $g$ of the ideal ...
1
vote
1answer
47 views

Multiplicative inverse of the power series $e^x - c$ for $c \neq 1$.

We know that the power series $f(x)= e^x -c \in \mathbb C[[x]]$ for $c \neq 1$, has a multiplicative inverse, since it's constant coefficient is non-zero. I was wondering whether the inverse is known ...
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0answers
22 views

Prime ideals in $C(X)$ and $C^*(X)$ and to be correspond

we know that every maximal ideal in $C(X)$ is in this form: $$M^p=\left\{\,f \in C^*(x):\ p\in cl_{\beta X} Z\left(f\right)\,\right\}$$ and every maximal ideal in $C^*(X)$ is ...
4
votes
1answer
33 views

Perfect number in gaussian integers

We have complete description about irreducibles in the ring Z[i],of gaussian integers. Now I was trying to define suitably the notion of "perfect number" in Z[i]. But the problem is unique ...
1
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0answers
13 views

Can We Use Extension of Scalars (Or Some Other Neat Way) to Prove This

Let $k$ be a field and $b(x)$ be a polynomial of degree at least $1$ in $k[x]$. Then for any given $f(x)\in k[x]$, there is an integer $m$ and polynomials $d_0(x), \ldots, d_m(x)\in k[x]$ such that ...
-3
votes
2answers
100 views

Prove that a finite Boolean ring is isomorphic to $\mathbb Z_2\times \mathbb Z_2\times\cdots\times \mathbb Z_2$ [duplicate]

If $R$ is a finite Boolean ring with $1\neq 0$ then prove that $R\cong \mathbb Z_2\times \mathbb Z_2\times \cdots\times \mathbb Z_2$. How should I proceed? Please give some hints only.
1
vote
2answers
62 views

Jacobson radical of $\mathbb{F}_{2}(t)[x]/(x^4-t^2)$

Let $\mathbb{F}_{2}$ be the field of two elements. Let $R=\mathbb{F}_{2}(t)[x]/(x^4-t^2)$. Why is $R/J(R)$ equal to $\mathbb{F}_{2}(t)[x]/(t-x^2)$? here $J(R)$ denotes the Jacobson radical of $R$.