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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Does a finite generated submodule with another element always generate a finite submodule?

Let M be a module and N be a finite generated submodule of M, and x$\in$M\N, then consider the existence of module B which is generated by N and x. I think, as N is f.g thus we can see N as ...
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1answer
35 views

Polynomial rings and quotients

Let $F$ be a field, $x$ an indeterminate in $F$, and $f(x)\in F[x]$ a polynomial with degree n. If the "Overline" denotes the canonical homomorphism from $F[x]\rightarrow F[x]/<f(x)>$, then, how ...
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1answer
43 views

Why is this Hilbert's Syzygy theorem?

In Lang's Algebra, chapter XXI, §4, on p. 861 he describes the standard construction of a graded (in principle infinite) free resolution of a finite graded module $M$ over the polynomial ring $A = ...
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1answer
27 views

Non-commutative noetherian integral domain-Ore condition

Let $R$ be a non-commutative integral domain with unity which is also a right Noetherian ring. By integral domain I mean that the product of nonzero elements is always nonzero. I am trying to show ...
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1answer
22 views

Concerning ideals of $\mathbb Z[\sqrt m]$ and $\mathbb Z[\sqrt m] [x] $

For a given integer $m<-1$ or non-square integer $m>1$ , how do we calculate the quotient ring $\mathbb Z[\sqrt m]/I$ , for example its order or whether it is a field or has zero divisors or not ...
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1answer
54 views

How to prove that the set of maximal elements of a set of prime ideals is finite

Let $A$ be a subset of ${\rm Spec}(R)$ with $R$ noetherian Are there any techniques to prove that ${\rm max}(A)$ (ie the set of maximal elements of $A$) is finite? I'm looking for equivalent ...
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1answer
46 views

Does validity of Bezout identity in integral domain implies the domain is PID ?

Let $D$ be an integral domain such that for any $a,b \in D$ , $Da+Db$ is a principal ideal , then must $D$ necessarily be a principal ideal domain i.e. should all the ideals of $D$ be principal ? ...
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1answer
28 views

Differences between primitive central idempotents and primitive orthogonal idempotents

I asked this question in mathoverflow. But it was closed. So I ask it here. If we have a complete set of primitive orthogonal idempotents of an algebra $A$, then we can obtain simple modules, ...
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4answers
69 views

Is $x^2 + xy + y^2$ irreducible in $\mathbb Q[x,y]$?

I guess I'm just not sure how to approach factorization outside of $\mathbb Q[x]$. I tried looking at it as a polynomial in $(\mathbb Q[x])[y]$, but the only tricks I know are Eisenstein's Criterion, ...
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1answer
28 views

Suppose that $|Z(R)| = 1$. Show that $|C_R(a)| \neq p$.

Let p be a fixed prime. Let $R$ any ring of order $p^2$ with identity. Suppose that $|Z(R)| = 1$, where $Z(R) = \{z \in R; zr = rz, \forall r \in R\}$ and $C_R(a) = \{r \in R; ra = ar\}$. Show that ...
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1answer
40 views

Finitely generated ideal containing non finitely generated ideal

I've been thinking about the following Rotman's excercise, and just can't find an answer: Give an example of a commutative ring $R$ containing proper ideals $I\subsetneq J\subsetneq R$ with $J$ ...
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1answer
29 views

Examples of commutative rings where the prime subring is not direct summand?

My question consists almost in the title. My motivation is the study of some tensor products $A\otimes_\mathbb{Z} B$. For a (commutative) ring, let us call prime subring the subring generated by $1$ ...
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2answers
43 views

A universal construction of the field of fractions of an integral domain?

Let $R$ be an integral domain and For a field $\hat R$ consider the following : There is an injective ring homomorphism $i:R \to \hat R$ such that for any field $F$ and any injective ring homomorphism ...
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1answer
66 views

Show that there exist a noncommutative ring (with identity) of order $p^3$.

Let p be a fixed prime. Show that there exist a noncommutative ring (with identity) of order $p^3$. RemarkI was able to $p = 2$: $U_n(\mathbb{Z}_2)$ - the set of $n \times n$ matrices with entries ...
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2answers
28 views

nilpotent right ideals

Theorem 3: Every nilpotent right (left) ideal is contained in a nilpotent two-sided ideal. Proof: Let $I$ be a nilpotent right ideal of $R$. By induction $(I + RI)^n ≤ I^n + RI^n$ for all ...
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2answers
21 views

A and B left ideals of ring R. Is $BA⊆A$?

Let $R$ be a ring. Let $A$ be left ideal of $R$, and $B$ be a left ideal of $R$. Is there any way I could show that $BA⊆A$? I was trying to use this fact to help me with another question, but I'm ...
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3answers
40 views

Prove $1+x$ is a unit in $R: \text{commutative ring}$ where $x$ is nilpotent [duplicate]

Prove $1+x$ is a unit in $R: \text{commutative ring}$ where $x$ is nilpotent do I need to make use of a Taylor series expansion for this? $(1+x)(1+x)^{-1} = 1 \implies (1+x)^{-1} = ...
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0answers
29 views

Unfamiliar w/ Ring Notation

I'm used to seeing rings represented as sets, but in one of my homework problems, I am asked to: Find the number of zero-divisors of $R_{x^2-x}$. Can somebody please explain what this notation ...
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1answer
52 views

Products and relationships of ideals of Ring R.

Let $R$ be a ring and let $I$ be a left ideal of $R$. (a) Let $K$ be a left ideal of $R$. Show that $(IK)^{n} \subseteq I^{n}K$ for all $n \in \mathbb{N}$ (b) Show that $I+ IR$ is a two-sided ideal ...
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1answer
48 views

Do partially ordered rings that aren't totally-ordered, which nonetheless satisfy $x+y \geq 0 \rightarrow x \geq 0 \vee y \geq 0,$ exist?

(All my rings are commutative and unital. I include $1 \geq 0$ as a partially-ordered ring axiom.) Let $R$ denote a partially-ordered ring. Observe that if $R$ is totally-ordered, then $R$ satisfies ...
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1answer
43 views

Question related to ring theory

Let $p$ be an odd prime and let $1 + \frac{1}{2} + \cdots + \frac 1{p-1} = \frac ab$, where $a,b$ are integers. Show that $p\mid a$. (Hint: As $a$ runs through $U_p$, so does $a^{-1}$.) P.S. $U_p$ ...
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0answers
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fundamental theorem of symmetric polynomials

Let $P_n(a,b,c)$ be a polynomial of variables $a,b,c$. By Newton's fundamental theorem of symmetric polynomials, there is a unique $P_n$ such that $$ x^n+y^n+z^n=P_n(x+y+z, x^2+y^2+z^2,x^3+y^3+z^3). ...
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1answer
20 views

If f is surjective then f is not a right divisor of zero

Let $R$ be a ring and $M$ a R-module. For $r\in R$ define $f:M\to M$ by $f(s)=sr$. Show that $f$ is injective if and only if $r$ is not a right zero divisor. I have done a similar problem to this in ...
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1answer
46 views

What is a linear combination, exactly?

I'm used to the definition of linear combination used in linear algebra textbooks. I'm reading the book Algebra by Artin and on page 357 he says: If $R$ is the ring $\mathbb{Z}[x]$ of integer ...
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2answers
51 views

Ring homomorphism takes discriminant to discriminant

Let $R[x] \xrightarrow{\sigma} S[x]$ be a ring homomorphism where $R,S$ are integral domains of characteristic $0$. Is it true that for any monic polynomial $f(x) \in ...
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1answer
46 views

Is $x \in X$ isolated iff the ideal $M_x := \{f \in C(X, \mathbb R) :f(x)=0\}$ of $C(X, \mathbb R)$ is principal ?

Let $(X,d)$ be a metric space , then is $x \in X$ isolated iff the ideal $M_x := \{f \in C(X, \mathbb R) :f(x)=0\}$ of $C(X, \mathbb R)$ is principal ? Do we need completeness of $X$ ?
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1answer
22 views

If $(X,d)$ is a finite metric space , then is every prime ideal of $C(X, \mathbb R)$ maximal?

If $(X,d)$ is a finite metric space , then is every prime ideal of $C(X, \mathbb R)$ maximal ? The thing is , since $X$ is finite , so it is compact , so ideal $M$ is maximal iff it is of the form ...
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1answer
21 views

How can I show that any one to one endomorphism of an Artinian module is an automorphism?

How can I show that any one to one endomorphism of an Artinian module $M$ is an automorphism? I was given this question and I presume that it is really to show that Artinian modules are co-hopfian. ...
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1answer
38 views

To find all integers $n > 1$ for which $(n-1)!$ is a zero-divisor in $Z_n$.

To find all integers $n > 1$ for which $(n-1)!$ is a zero-divisor in $Z_n$. (Gallian Problem) $Z_n$ does not contain any zero divisors when $n$ is a prime number. So we look at the composite ...
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2answers
51 views

Why Can't we Factor Invertible Elements?

I'm currently studying Herstein's Algebra; specifically, UFDs and the abstract notion of factorization. This is perhaps more of an intuitive question than one with a hard answer. We define ...
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1answer
103 views

What does $\Bbb{Z}/(2) \times \Bbb{Z}/(3) \times \dots$ do?

Take the product of rings $M = \Bbb{Z}/(2) \times \Bbb{Z}/(3) \times \dots$ over the primes or in general take any infinite set of quotient modules of a ring $R$ and form their product. It's true ...
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1answer
49 views

Ring of Infinitely Differentiable Functions

Denote the ring $\text{C}^{\infty}$ i.e. infinitely differentiable functions from $\mathbb{R}\mapsto\mathbb{R}$. I have managed to prove that this ring is a Principal Ideal Domain. I must also prove ...
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1answer
34 views

Polynomial ring with integral coefficients is integral

Let $B$ be a ring and $A\subset B$ a subring. Assume that $B$ is integral over $A$. I have to prove that $B[X]$ is integral over $A[X]$. I tried writing down an integral relation for $f(X)\in B[X]$ ...
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2answers
25 views

Showing that an element is prime in $\mathbb{Z}$[i]

Let p be a prime integer, and suppose p = a2 + b2 has NO integer solution. The exercise asks that if p = a2 + b2 has no solution, then p is a prime in the set of Gaussian integers $\mathbb{Z}$[i], ...
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1answer
37 views

Invertible element vs zero divisor in a ring

Let $R$ be a ring and $x,y\in R$ such that $yx=1$ and $xy\ne 1$. Prove there is $z\ne 0$ such that $yz=zx=0$. My first thought is that y is not an invertible element. Does that mean that it is a ...
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2answers
68 views

Show that $End_{\mathbb{K}}(\mathbb{V})$ is Dedekind finite ring.

A ring $R$ is said to be Dedekind finite if $ab=1 \Rightarrow ba=1$. Let $\mathbb{V}$ a finite-dimensional $\mathbb{K}$-vector space, show that $End_{\mathbb{K}}(\mathbb{V})$ is Dedekind finite ring.
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1answer
63 views

Given two idempotents $a,b \in R$ such that $a+b$ is idempotent then $a$ and $b$ commute.

Let $R$ be a ring with identity. An element $a \in R$ it is idempotent if $a^2=a$. Show that given two idempotents $a,b \in R$ such that $a+b$ is idempotent then $a$ and $b$ commute. Remark: ...
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1answer
59 views

Construction of homomorphism between $^\ast\mathbb{R}$ and $^*\mathbb{Q\cap L}$

Denote by $\mathbb{I}$ the ring of infinitesimals and by $\mathbb{L}$ the ring of finite hyper-reals. Prove that $$\mathbb{R}\cong{^\ast\mathbb{Q\cap L/^\ast Q\cap I}}.$$ I thought using the ...
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1answer
18 views

A question concerning to show that $V(I)$ is open if $I$ is radical ideal

Let $I:=(f_1 ,...,f_k)$ be a finitely generated ideal of $C(X,\mathbb R)$ such that $\mathrm{rad}(I)=I$, $f:=\sqrt{\sum_{m=1}^k |f_m|}=\sum _{m=1}^k g_mf_m$ where $g_i $'s are real valued ...
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1answer
31 views

To show that an integer $m$ is prime element in $\Bbb Z[i]$ if $m$ is a prime number of the form $4n+3$.

Let $p$ be a prime number of the form $4n+1$. Then show that $p = a^2 + b^2$ for some $a,b \in \Bbb Z$ and $p$ is not prime in $\Bbb Z[i]$. Also show that an integer $m$ is prime element in $\Bbb ...
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0answers
20 views

Tensor product of algebras and generating sets.

Let $A$ be a module over $k$ generated by $x$ and $y$. The generating set for $A \otimes_k A$ is $\{x \otimes x, x \otimes y, y \otimes x, y \otimes y\}$. But does this still hold if $A$ is an algebra ...
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0answers
24 views

In a Euclidean domain, show that for nonzero $a,b \in D$, $v(a) < v(ab)$ iff $b$ is not a unit of D.

The function $v(x)$ is a Euclidean function on an integral domain, D. Proof : Suppose that $v(a) < v(ab)$. If $b$ were a unit, then $a$ and $ab$ would be associates. We have $a = abu$ and $ab = ...
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An essential right ideal in a ring

Let $S⊆R$ be rings with unity such that $S_S$ is essential in $R_S$. If $r∈R$ is a nonzero element there exists an $s_0∈S$ with $rs_0$ a nonzero element of $S$. Now, could we find a right ideal $I$ ...
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2answers
34 views

Polynomial factorisation for unique factor domain

Suppose $R$ is a UFD and $f \in R[X]$ such that $\deg f > 0$ and $f$ has a root $\alpha \in R$. Show that $f = (X - \alpha) g$ for some $g \in R[X]$. (Suggestion: Write $f = a_0 + a_1 X + \dotsc ...
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1answer
26 views

Proving that a surjective homomorphism can help generate a finitely generated k-algebra

I am trying to understand a proof in Dummit and Foote. It can be found in Chapter 15, Section 1, Corollary 5. The corollary is The ring $R$ is a finitely generated $k$-algebra iff there is some ...
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2answers
62 views

$I^{j}/I^{j+1} \cong R/I$ for any ideal I in ring R.

Let $R$ be a commutative ring with $1$ and $I$ be an ideal in it. Let $\overline{\alpha} \in I^j\setminus I^{j+1}$ and define $\theta\colon R \to I^j/I^{j+1}$ by $\theta(x)=\overline{\alpha x}$. My ...
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0answers
35 views

Show that if $a \in R$ has more than one left-inverse then it has infinite. [duplicate]

Let R is a ring with identity $1$. Show that if $a \in R$ has more than one left-inverse then it has infinite.
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0answers
45 views

Ring homomorphisms on the set of rationals that coincide on integers

Let $R$ be a ring and let $f, g: \mathbb Q \to R$ be two ring homomorphisms such that $f|_{\mathbb Z}=g|_{\mathbb Z}.$ Then $f=g.$ I was trying to prove the above mentioned statement. According ...
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0answers
5 views

Proof that ideal in Lie ring can be represented as sum of 2 Lie subrings.

Let $K$ be a commutative ring and $m≥3$. Let $L(m,K)$ be a Lie subring of matrices with coefficients from ring $K$ that contains matrices with null traces, $L(m,K)={(a_{ij})∈M_m(K)|\sum\limits_{i = ...
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1answer
44 views

Every nilpotent left ideal is contained in a nilpotent 2 sided ideal.

Question: Let $R$ be a ring and let $J$ be a left ideal of $R$. Assume that $J$ is nilpotent. Prove that $J$ is contained in a nilpotent 2-sided ideal of $R$. Comments: I have found lots of ...