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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Let $R$ be a commutative ring with $1 \ne 0$, and let $0 \ne e \in R$ be an idempotent element. Prove the following:

Let $R$ be a commutative ring with $1 \ne 0$, and let $0 \ne e \in R$ be an idempotent element. Note that $eR=\{er|r \in R\}$ is also a commutative ring with identity element $e$. (1) If I is an ...
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28 views

Let ideal $I$ be generated by polynomial $p(x)$ in the ring $F[x]$ F-field.

Let ideal $I$ be generated by polynomial $p(x)$ in the ring $F[x]$ F-field. Prove: that the polynomials f(x), g(x) are in the same factor class of the ring $\implies f(x)=g(x)(mod\ p(x))$ ...
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Let $R = \mathbb{Z} + x\mathbb{Q}[x]$. Find all the irreducibles in $R$.

Let $R = \mathbb{Z} + x\mathbb{Q}[x] \subset \mathbb{Q}[x]$. Find the irreducibles of $R$. Show that the irreducible elements in $R$ are $\pm p$ for prime integers $p$ and the irreducible ...
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45 views

Form of maximal ideals in an algebraicaly closed polynomial ring

I have been trying to prove the following bijection which is a consequence of the nullstellansatz $$\{\text{maximal ideals of }\mathbb{C}[x_1,\dots,x_n] \} \leftrightarrow \{\text{points in ...
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35 views

Every polynomial has a root

Let $A$ be a commutatif ring, and $f\in A[T]$ une polynome. Then in the $A$-algebre $B=A[T]/(f)$ the polynomial $f$ has a root, namely $T \mod (f)$, because $f(T)\mod (f)=f(T)\mod (f)=0$. Do you ...
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35 views

If in a UFD every maximal ideal is principal then it is a PID

I want to prove that if in a UFD every maximal ideal is principal then it is a PID. My line of attack is: If it is a field i.e. it has no non-zero proper ideal, then we are done. Otherwise ...
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20 views

a factorisation of $x^9−x$ into a product of irreducibles in $\mathbf F_3$

Firstly, given the degree of the extension $[\mathbf F_9 : \mathbf F_3]$ is 2. Then, need to write it as a product of irreducible polynomial. What I've got was that $x(x+1)(x-1)(x^2+1)(x^4+1)$ ...
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An example of an ideal of order $12$

Provide an example of an ideal in $R=\mathbb{Z}_6\times\mathbb{Z}_{10}$ that has order $12$, and indicate whether the ideal is a principal ideal (if it is, then identify the generator for the ...
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1answer
40 views

Definition of Local Ring

I was reading some article on local rings, and it gave the following (equivalent) definitions: A ring $R$ is $local$ if it satisfies one of the following equivalent properties (so this needs proving) ...
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Ideals of $\mathbb{Z}_6\times\mathbb{Z}_{10}$

Two things concerning the ring $R=\mathbb{Z}_6\times\mathbb{Z}_{10}$: Identify a subset of $R$ that is a subring of $R$ but not an ideal of $R$. Identify a maximal ideal in $R$. For the first ...
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42 views

Rings isomorphic to $\mathbb{Z}_6\times\mathbb{Z}_{10}$

What are five ring properties that hold for each ring that is isomorphic to $R=\mathbb{Z}_6\times\mathbb{Z}_{10}$, but not for every ring? Suppose $Q\approx R$. Then $Q$ has unity, $Q$ is not a ...
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115 views

Let $R$ be a Euclidean domain with degree function $\varphi$ and $R$ not a field. If $a \ne 0$ and $b \ne 0$ be two elements in $R$…

Let $R$ be a Euclidean domain with degree function $\varphi$ and $R$ is not a field. Prove the following: (1) Let $a \ne 0$ and $b \ne 0$ be two elements in $R$. Suppose that $a\mid b$ and $b \nmid ...
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1answer
100 views

Is it true that $R^n\simeq R^m$ as rings implies $m=n$?

Let $R$ be a commutative ring. We know that if $R^n\simeq R^m$ as $R$-modules for some positive integers $n,m$ then $n=m$. But is it still true when they are isomorphic as rings? Thanks!
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1answer
120 views

Corollary of Gauss's Lemma (polynomials)

I am trying to prove the following result. I have outlined my attempt at a proof but I get stuck. Any help would be welcome! Theorem: Let $R$ be a UFD and let $K$ be its field of fractions. ...
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Proving the existence of a square root of $ -1_{A} $ in a $ 2 $-dimensional unital algebra $ A $ over $ \Bbb{R} $.

Suppose that $ A $ is a $ 2 $-dimensional unital algebra over $ \Bbb{R} $ with a basis $ \{ 1_{A},u \} $, and assume that $ A $ does not have any zero divisors. Show that $ A $ contains an element $ b ...
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108 views

Finite commutative ring with more than $\frac{2}{3}$ of its elements idempotent

Suppose that $R$ is a finite commutative ring with identity element, such that more than $\frac{2}{3}$ of elements are idempotent. Prove that all of elements are idempotent. Please give me a ...
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1answer
28 views

Construction of the discrete valuation ring

Let $K$ be a field. A surjective transformation: $v: K \to \mathbb{Z}\cup\{\infty\}$ is defined as a discrete valuation, if for any $a, b \in K$, the following statements hold true: $v(ab) = v(a) + ...
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59 views

Characterize all finite unital rings with only zero divisors

Is it true that for every finite (for simplicity, commutative) ring $R$ in which every element not equal to $1$ is a zero divisor, is isomorphic to the zero ring or $\mathbb{Z}/2\mathbb{Z}$, ...
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19 views

What is the number of elements of $\mathbb Z[i] /I $, where $I:=\{a+ib \in \mathbb Z[i] : 2 \mid a-b\}$?

I know that $I:=\{a+ib \in \mathbb Z[i] : 2\mid a-b\}$ is a maximal ideal of $\mathbb Z[i]$. My question is: what is the number of elements of $\mathbb Z[i] /I $? I am totally stuck. Please help ...
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74 views

Show that if $x ≡ 1 (\text{mod } λ)$, then $x^3 ≡ 1 (\text{mod } λ^3)$…

Let $λ = (3 + \sqrt{−3})/2 ∈ Q[\sqrt{−3}]$. How do I show that if $x ≡ 1 (\text{mod } λ)$, then $x^3 ≡ 1 (\text{mod } λ^3)$. Also, how do I show that if $x ≡ −1(\text{mod } λ)$, then $x^3 ≡ −1 ...
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Isomorphism inbetween a factor ring and Cartesian product of factor rings [duplicate]

Let R be principal ideal ring and $a_1, ..., a_n \in R$, with $gcf(a_i, a_j) = 1$, for all $i, j \in \{1, ..., n\}$ (with $gcf$ being the greatest common factor of $a_i, a_j$). Show that the ...
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51 views

How to show fraction field is flat (without localization)

Here I asked that if one can prove the field of fraction of a domain is flat. The answers used localization, which I am not familiar with. Can anyone prove it without using localization?
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Why field of fractions is flat?

I want to show this lemma: Let $R$ be a domain. If $A$ is a torsion $R$-module, then $\operatorname{Tor}_1^R (K,A)\cong A$ where $\operatorname{Frac}(R)=Q$ and $K=Q/R$. When I was reading ...
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Please help collecting examples of finite/infinite rings satisfying different conditions about units/zero divisors (Added question 4)

0) Every nonzero element of a finite ring is either a zero divisor or a unit. This is proved in Every nonzero element in a finite ring is either a unit or a zero divisor 1) If a ring R satisfies the ...
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Quotients of $\mathbb{Z}[i]$

Let $\mathbb{Z}[i]$ be the ring of Gauss integers. For a simple representation it is all the complex numbers of the form $a+ib$ such that $a,b \in \mathbb{Z}$. It is known that $\mathbb{Z}[i]$ is a ...
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Pre-image of a ring homomorphism

Let $\phi:R \rightarrow S$ be a ring homomorphism and let $J$ be an ideal in $S$, then it is quite easy to prove that $\phi^{-1}(J)$ is an ideal in $R$. But can someone help me with proving the ...
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Product of rings: If K is an ideal of RxS, then there exists I ideal of R, J ideal of S such that K=IxJ.

Let R and S be two rings. We consider the product R×S. It is a ring with operations of sum and product defined coordinate by coordinate, i.e.: (r1, s1) + (r2, s2) = (r1 + r2, s1 + s2) and (r1, s1) · ...
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Characterization of epic morphisms in the category of rings.

Let $C$ denotes the category of rings (with identities). It's easy to prove that for any $R\in \text{ob }C$, any multiplicative set $S\subset R$ and any ideal $I\leqslant R$, both $R\to S^{-1}R$ and ...
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1answer
56 views

Injection and surjection over free modules.

Let $A$ be a commutative ring and $M$ an $A$-module. Suppose to have both an injection $A^s \to M$ and a surjection $A^s \to M$ of module homomorphisms. Show that $M \simeq A^s$. This point is ...
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1answer
49 views

If $R/I$ and $R/J$ are semisimple, then so is $R/I\cap J$.

Let $R$ be a not necessarily commutative ring. If $I$ and $J$ are (two-sided) ideals in $R$ such that $R/I$ and $R/J$ are both semi-simple rings, then so is $R/I\cap J$. I tried the following: ...
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1answer
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Matrix Ring of a Semisimple Ring

I recently read the concept of semi-simplicity of a (not necessarily commutative) ring. A ring $R$ is said to be semi-simple if $R$ as a left module over itself is a semi-simple module (This in turn ...
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Is $3$ a prime element of $\mathbb{Z[\eta]}$?

How to check whether $3$ is a prime element or not in $\mathbb{Z[\eta]}$, where $\eta$ is a $17$th primitive root of unity. Also in general how can we check an element is prime or not in ...
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Maximal ideals of polynomial ring $\mathbb Z_n[x]$

How would I find one? say $n=p^2q^2$ for $p,q$ primes
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51 views

Group of units in the rings $\mathbb I_9 $ and $\mathbb I_{15}$?

The question I need help is: Prove that $U(\mathbb I_9) \cong \mathbb I_6$ and $U(\mathbb I_{15}) \cong \mathbb I_4 \times \mathbb I_2$. U() is the group of units in a ring All the "I" are ...
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28 views

Show that the Gauss ring is Euclidean ring

Show that the Gauss ring $ (\mathbb{Z}[i]=\{a+bi \mid a,b \in \mathbb{Z}\},+,\cdot ) $ is Euclidean ring with a norm $d(a+bi)=a^{2}+b^{2}$ How to prove that theorem? I started checking the first ...
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1answer
18 views

Are the isomorphism classes of simple left ideals in a semisimple ring finite?

Suppose $R$ is a unital semi simple ring, not necessarily commutative. It's known that there are only finitely many isomorphism classes of simple left ideals. Are these isomorphism classes ...
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34 views

Non-unique factorization in $\mathbb{Z}[\sqrt{-5}]$

I want to show that the decomposition into irreducible factors in the ring $$\mathbb{Z}[\sqrt{-5}] = \{a + b\sqrt{-5}|\space a, b \in \mathbb{Z}\}$$ is not unique, except for the order of factors ...
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$R/I^n$ is a local ring

Let $I$ be a two sided ideal of a ring $R$ such that $I$ is maximal as a right ideal. I need to show that $R/I^n$ is a local ring, for every $n \geqslant 1$. For $n=1$ I was able to show that the ...
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Ring and sum of idempotent elements

Let $R$ be a ring with identity which for every $x\in R$ there exist two idempotent elements $e_1,e_2$ such that $x=e_1+e_2$ and $e_1e_2=e_2e_1$. Prove that: $x^3=x$ for every $x\in R$.
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zero element in tensor product of a localization ring and a module

Let $R$ be a commutative ring with $1$. Let $f$ be a non-nilpotent element of $R$ and let $R_f$ be a localization of $R$ by the multiplicative set $\{ f^i \mid i=0,1,2,\dots\}$. Let $M$ be an ...
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1answer
29 views

If $R$ is a Boolean ring with $\mid R \mid > 2$, determine all the zero divisors of $R$.

If $R$ is a Boolean ring with $\mid R \mid > 2$, determine all the zero divisors of $R$. My attempt: Let $a, b\in R$, $a \neq 0$ and $ab = 0$. How do I prove that $b = 0$?
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36 views

Show that there exists an idempotent element such that $Ra=Re$ holds for ring $R$

Let $R$ be a ring with 1 such that for every element $x$ in $R$, $\exists y\in R$ such that $xyx=x$ holds. Show that for any $a\in R \exists$ idempotent $e\in R$ such that $Ra=Re$ Let $a=aya$. ...
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Intersection of ideals generated by two relatively prime elements

I am wondering how to prove the following statement: Let $R$ be a PID, $a,b$ are relatively prime. Then $\langle a\rangle \cap \langle b\rangle = \langle ab\rangle$ Progress: I think it ...
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Finding the left (or right) ideals of the ring of $n\times n$ matrices

Just give me a hint, since this is assessment! DO NOT TELL ME THE IDEAL I want to find the left (or right) ideals of the ring of $n\times n$ complex valued matrices. Now the definition is (for ...
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1answer
34 views

Questions about the formal derivative over $F[x]$

Let $F$ be a commutative ring and $f(x)=a_{0}+a_{1}x+.......+a_{n}x^n$ be in $F[x]$. Define $f'(x)=a_{1}+2a_{2}x+...+na_{n}x^{n-1}$ to be derivative of $f(x)$. Prove that $(f+g)'(x)=f'(x)+g'(x)$, ...
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1answer
31 views

Contradiction in proof that in an integral domain, every prime is irreducible.

Let $\pi$ be a prime element in an integral domain. So, $\pi$ is a non-unit and if $\pi \mid ab \ $ then $\pi \mid a$ or $\pi \mid b$. An irreducible element $z$ is an element such that if $z=ab$, ...
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3answers
49 views

How many elements are in the quotient ring $\frac{\mathbb Z_3[x]}{\langle 2x^3+ x+1\rangle} $

How many elements are in the quotient ring $\displaystyle \frac{\mathbb Z_3[x]}{\langle 2x^3+ x+1\rangle}$ ? I guess I should be using the division algorithm but I'm stuck on how to figure it out.
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$\varphi : R → S$ is a epimorphism from $R$ to ring $S$, let $I$ be an ideal of $R$. Prove $\varphi (I) = S$ if and only if $R = I +Ker(\varphi)$

Let $\varphi : R → S$ be an epimorphism from ring $R$ to ring $S$, and let $I$ be an ideal of $R$. Prove that $\varphi (I) = S$ if and only if $R = I +Ker(\varphi)$ I am quite confused on what ...
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1answer
53 views

Find the smallest subring of $\mathbb{R}$ containing $\frac 12$.

Find the smallest subring of $\mathbb{R}$ containing $\frac 12$. My attempt: I have formed a subring containing $\frac 12$ i.e. $\{\frac n2 | n \in \mathbb{Z}\}\cup\{(\frac 12)^{k}|k\in\mathbb{Z}, ...
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53 views

Find the smallest subring of $\mathbb{Z}$ containing $8$.

Find the smallest subring of $\mathbb{Z}$ containing $8$. My attempt: I have formed a subring containing 8 i.e. $\{8n \mid n \in \mathbb{Z}\}\cup\{8^k \mid k\in\mathbb{Z}, k>0\}$. But how do I know ...