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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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 $1$ i....
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79 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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83 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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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 $n≥...
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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
82 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} = \displaystyle\...
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0answers
45 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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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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59 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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49 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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1answer
22 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
94 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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66 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 R[x],\sigma(disc(f(x)))=disc(\...
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1answer
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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
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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
31 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
175 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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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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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
168 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
47 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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35 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
154 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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79 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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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: I'...
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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 first ...
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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 continuous ...
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1answer
58 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 $\Bbb Z[i]$...
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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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1answer
101 views

Why are $a$ and $ab$ associates if $b$ is a unit?

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 =...
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40 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
30 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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71 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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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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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 to ...
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1answer
61 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 ...
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1answer
108 views

Irreducible elements in $\mathbb{Z} [ \sqrt{-3}]$

I am having trouble this problem in my textbook. It ask us to determine whether the following elements are irreducible in $\mathbb{Z} [ \sqrt{-3} ] = \{a+b\sqrt{-3} : a,b \in \mathbb{Z}\}.$ $\sqrt{-3}...
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Quaternion algebra of characteristic 2?

I've been reading up on quaternion algebras recently and noticed the vast majority of theorems are contingent on setting the characteristic $p \neq 2$. In particular, this is true for the central ...
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Why Jacobson, but not the left (right) maximals individually?

When we are working with Path Algebras, it does not need very sophisticated tools to prove that for a finite, connected, acyclic quiver $Q$, the Jacobson Radical of $KQ$ is nothing but the arrow ideal....
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Irreducible elements and maximal ideals

Let $A$ be some ring. It is well known that if $A$ is an integral domain, then an element $x$ is irreducible iff the principal ideal $xA$ is maximal in the set of principal ideals of $A$. Questions: ...
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Looking for at least one surjective ring homomorphism from $M_n(R)$ to $R$

Let $R$ be a ring , I am looking for a surjective ring homomporphism from $M_n(R)$ to $R$ . Please help . Thanks in advance .
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319 views

Pull back image of maximal ideal under surjective ring homomorphism is maximal

Let $f :R \to S$ be a surjective ring homomorphism , $M$ be a maximal ideal of $S$ , I am writing a proof showing $f^{-1}(M)$ is a maximal ideal of $R$ , Please verify whether it is correct or not . ...
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For $F[\alpha]$ as a finite field extension of $F$, is $1/\alpha \in F[\alpha]$?

Is the multiplicative inverse of $\alpha$, that is, (1/$\alpha$) contained within $F[\alpha]$? Thanks in advance.
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1answer
28 views

Is there a semiprime ring (with 1) with non-zero singular ideal?

I study semiprime rings and modules and I haven't found an example of a semiprime ring with non-zero singular ideal nor I haven't been able to prove that every semiprime ring is nonsingular.
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Localization in Noetherian rings

Let $R$ be a commutative noetherian ring with no nonzero nilpotents. Is every localization of $R$ at a maximal ideal a field?
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55 views

Non-Noetherian ring with Noetherian quotient

Let $R$ be a commutative ring with $1 \ne 0$. I'm trying to prove that if $R$ contains an ideal $I$ that is not finitely generated, then $R$ contains a proper ideal $J$ such that $R/J$ is Noetherian....
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Trying to prove $Z(I(A))=\bar A$

For every ideal $I$ of $C[0,1]$ , define $Z(I):=\{x \in [0,1] :f(x)=0 , \forall f \in I\}$ and for every $A \subseteq [0,1]$ , let $I(A):=\{f \in C[0,1] : f(x)=0 , \forall x \in A\}$ . Then ...
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1answer
106 views

Analysis of the ideals of $C[0,1]$

For every ideal $I$ of $C[0,1]$ , define $Z(I):=\{x \in [0,1] :f(x)=0 , \forall f \in I\}$ and for every $A \subseteq [0,1]$ , let $I(A):=\{f \in C[0,1] : f(x)=0 , \forall x \in A\}$ . Then ...
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32 views

prove that the quotient ring S3/T3 is isomorphic to D3

Could you please help with this question? I've already shown that T_3 is an ideal of S_3. Thanks,
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2answers
25 views

Proof in ring homomorphism

I am looking at the solution one my teacher gave me to prove that a function $\phi$ from a ring R to a another ring S is a ring homomorphism. He checks: $$\phi(x+y) = \phi(x) + \phi(y) \\ \phi(x\...