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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2
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1answer
46 views

How to go about proving that $-1+(x-4)(x-3)(x-2)(x-1)$ is irreducible in $\mathbb{Q}$?

How do you show that $-1+(x-4)(x-3)(x-2)(x-1)$ is irreducible in $\mathbb{Q}$? I don't think you can use the eisenstein criterion here
0
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2answers
28 views

Why is not the polynomial ring $R[x]$ a unique factorization domain, where $R$ is the quadratic integer ring $\mathbb{Z}[2\sqrt{2}]$?

Why is not the polynomial ring $R[x]$ a unique factorization domain, where $R$ is the quadratic integer ring $\mathbb{Z}[2\sqrt{2}]$? I'm trying to find a irreducible nonprime element or something but ...
0
votes
1answer
29 views

Notation meaning:$ R(x^2 + x +1)$

I'm doing a problem sheet on about rings and ideals, and would appreciate clarification on some notation. The problem is: "Let $R=\mathbb{Z}_2[x]$, the ring of polynomials with coefficients in ...
0
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1answer
28 views

$f$ be a ring automorphism on $R[x]$ such that $f(u)=u , \forall u \in R$ , then is it true that $f(x)=ax+b $ for some $a,b \in R$?

Let $R$ be a ring and $f:R[x] \to R[x]$ be a ring automorphism such that $f(u)=u , \forall u \in R$ , then is it true that $f(x)=ax+b $ for some $a,b \in R$ ?
1
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1answer
26 views

Existence of Divisor-Zeros in $c(X,\mathbb{R})$

Consider any topological space $X$ and $\mathbb{R}$ be with usual topology. The set of all continuous functions from $X$ to $\mathbb{R}$, denoted by $C(X,\mathbb{R})$, is a commutative ring with unity ...
0
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0answers
22 views

Let R be a commutative ring and let I,J be ideals of R.

Define $I \cap J = \{a \in R: a \in I \text{ and } a \in J\}$ $I+J = \{a+b \in R: a \in I, b\in J\}$ Suppose $R = \Bbb Z$ or $F[x],\ I = \langle a\rangle$ and $J = \langle b\rangle$ Identify $I ...
0
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1answer
13 views

Application of extended euclidean algorithm to find the inverse of polynomial

I'm trying to understand the Example. ( A quotient ring which is a field) on this ...
0
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0answers
19 views

Ring Theory Questions [duplicate]

To show:Given a ring R, show that there exist a ring R' with unity such that R is a subring (upto isomorphism) of it.
1
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1answer
31 views

Basic open sets in the Zariski topology are also compact.

Let $A$ be a commutative ring and $X = \text{Spec}(A)$. The closed sets are those of the form $V(E) = \{$ prime ideals $\hat{p} \subset A $ containing $E \}$. And the open sets are the complements ...
4
votes
1answer
61 views

$1+ab$ is a unit if and only if $1+ba$ is a unit. [duplicate]

Let $R$ be a ring with identity and $a,b \in R$ then prove that $1+ab$ is a unit if and only if $1+ba$ is a unit and find the inverse. Then there exist an element say $s \in R$ such that $(1+ab)s =1$ ...
1
vote
1answer
30 views

What is the dimension of $\mathbb R[x] / \langle x^3-x\rangle$ as a vector space over $\mathbb R$ ?

What is the dimension of $\mathbb R[x] / \langle x^3-x\rangle$ as a vector space over $\mathbb R$ ? Can someone please give some links , articles where I can study about polynomila rings and its ...
2
votes
1answer
25 views

To show that either R is a field or R is a finite ring with prime number of elements and $ab = 0$ for all $a,b \in R$.

Let $R$ be a commutative ring such that $R$ has no nontrivial ideal. Then show that either R is a field or R is a finite ring with prime number of elements and $ab = 0$ for all $a,b \in R$. I am ...
1
vote
1answer
36 views

Is $\mathbb R[x]/\langle (x-a)^2 \rangle $ isomorphic with some familiar ring structure ( where $a$ is a real number )?

Is $\mathbb R[x]/\langle (x-a)^2 \rangle $ isomorphic with some known ring ( where $a$ is a real number ) ? In particular is $\mathbb R [x] / \langle (x-1)^2 \rangle$ isomorphic with some known ring ? ...
1
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3answers
28 views

The characteristics of a subfield of a field is same as that of the field.

How to show that the characteristics of a subfield of a field is same as that of the field??
0
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3answers
37 views

To show that the field $\Bbb Z_p$ has no proper subfield when $p$ is a prime number.

To show that the field $\Bbb Z_p$ has no proper subfield when $p$ is a prime number.Is the reason of having no primes due to non factorization of $p$??
0
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1answer
34 views

Suppose that $R$ is a one-dimensional normal Noetherian local ring. Then the maximal ideal $m_R$ is principal

Theorem 11.2 (Matsumura's Commutative Ring theory) gives us equivalent conditions for a ring $R$ to be considered a DVR. I was stuck while reading the proof of $(4) \implies (3)$, (3) $R$ is a ...
1
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0answers
17 views

Tidy way to represent XOR over the ring of $2^{32} - 1$

I was reading about a cipher called Speck, which defines a system of equations using Addition Mod $2^{32}$ ($\boxplus$), Bit Rotation, and XOR. If we pretend that the additions were taken over ...
3
votes
1answer
47 views

Let $T = \{\frac ab \in \Bbb Q \mid \text{$a$ and $b$ are relatively prime and $5 \nmid b$}\}$

Let $T = \{\frac ab \in \Bbb Q \mid \text{$a$ and $b$ are relatively prime and $5 \nmid b$}\}$ . Show that $T$ is a ring under usual addition and multiplication. Also prove that $I = \{\frac ab \in T ...
1
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0answers
29 views

Intersection of modules is equal to product.

If $B$ is a commutative ring and let $\mathcal{Q}_1,\ldots,\mathcal{Q}_n$ ideals relative primes. Let $M$ be a $R$-module. I don't sure if this is true. Then $$(\mathcal{Q}_1\cap\cdots ...
1
vote
1answer
31 views

Why is an Ideal a subset of Rad(I)

Why is $I \subseteq Rad(I)$? I don't quite get this.
1
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1answer
14 views

Let D be a UFD. Show that a nonconstant divisor of a primitive polynomial in D[x] is again a primitive polynomial.

Laying out the clues, I have $f(x), g(x) \in D[x]$. Assume that $g(x)|f(x) = c(x)$ such that $f(x) = c(x)g(x)$. Given that g(x) is a primitive polynomial, does this mean any factorization such that ...
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2answers
35 views

Subring of artinian unital ring.

Is there any unital artinian ring such that it have a unital subring isomorphic to itself?
2
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1answer
29 views

If $I$ is a proper ideal of $C[0,1]$ , then should there exist $a \in [0,1]$ such that $f(a)=0 , \forall f \in I$?

Let $C[0,1]$ be the ring of all real valued continuous functions under point-wise addition and multiplication . We know that for every $a \in [0,1]$ , $\{f \in C[0,1] : f(a)=0\}$ is a proper ideal of ...
3
votes
1answer
48 views

In a finite ring $R$ with identity show that $ab =1$ implies $ba = 1$, where $a,b \in R$. [duplicate]

In a finite ring $R$ with identity show that $ab =1$ implies $ba = 1$, where $a,b \in R$. I am having difficulty in doing this since there is no condition that there is no zero divisors and how will ...
1
vote
2answers
27 views

Why are irreducible elements non-units?

I know this may seem trivial but I'm trying to grasp why irreducible elements are non-units. If an element p is a unit and b is its inverse, then $pb = 1, \forall p,b \in R$, R is a ring. Does this ...
1
vote
1answer
47 views

Zeros in the polynomial ring $\mathbb{R} [X,Y]$

I know that for $p(X) \in \mathbb{R} [X]$, $a$ is a zero of $p(X) \iff (X-a)|p(X)$. But what would the statement be for $p(X) \in \mathbb{R}[X,Y]$? This question comes from an example in my ...
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3answers
31 views

Prove the Ring Homomorphism is Surjective

Prove the following homomorphism is surjective. $$f : Z → \frac{Z}{3Z} × \frac{Z}{5Z}$$ I completely get the questions and i can prove it by working out a corresponding pre-image for all of ...
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0answers
7 views

How to calculate the thickness of a torus when volume & ID are known? [on hold]

I have a Torus with an ID of 16 mm & thickness of 2.1 mm. I am able to calculate the volume to 196.95 mm^3. Using this static volume, how am I able to calculate the thickness of the torus when the ...
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0answers
78 views

How do I prove that an ordered field and positive cone are bijective?

A field $K$ together with a binary relation $<$ is an ordered field provided the following hold: (O$0$) If $a \in K$, then one and only one of the following holds: $0<a$, $a=0$, or $a<0$. ...
4
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1answer
33 views

Determining the multiplicative group of a ring of polynomials

Let us say that we have the polynomial ring R[x]. Would it be possible to determine the order of the multiplicative group of R[x] modulo a polynomial f?
3
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1answer
30 views

Semisimplicity of a ring

As it is well-known, a ring with unity $R$ is semisimple if and only if each left $R$-module is projective. My question: Is simisimplicity of $R$ equivalent to each "simple" left $R$-module being ...
1
vote
1answer
39 views

Showing $\sqrt{N}$ is an ideal of $R$

I wish to show that if $N$ is an ideal of a commutitive ring $R$, then $$\sqrt{N} = \{a \in R \mid \exists k \in \mathbb{N}^{*} \textrm{s.t. }a^k \in N\}$$ is an ideal of $R$. Is this sufficient: " ...
0
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1answer
19 views

The dual algebra and linear functionals.

Suppose that we have a finite-dimensional algebra $A$ over $k$ generated by $\{x_1, x_2\}$. Then the dual algebra $A^*$ is generated by $\{x_1^*, x_2^*\}$ where $$x_1^* : A \rightarrow k \space ...
1
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3answers
35 views

Suppose that $R$ is a commutative ring with identity in which $1=0$. Show that $R=\{0\}$ [on hold]

Suppose that R is a commutative ring with identity in which $1=0$. Show that $R=\{0\}$.
2
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1answer
29 views

Proof of Unique factorization in Dedekind Rings .

Proof de unique factorizaation in Dedekind Rings. Algebraic Number fields, Janusz, Second edition. In the above proof, Theorem 3.13. Why of the corolary 3.7, ...
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2answers
62 views

How to show that $J$ is a left ideal of $R$?

Let $R$ be a ring and $x \in R$. Let, $J = \{ax|a \in R\}$ Show that $J$ is a left ideal of $R$. To show something is a left Ideal(or a right ideal), do we need to show that it is a subgroup first? ...
0
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0answers
17 views

Is the direction of containment right?

If $N$ is a submodule of $M$, then for any $x$ such that $xM = 0,$ then given arbitrary $n ∈ N, n ∈ M, xn = 0.$ Then the collection of $x$'s that annihilate $M$ contain the collection of $x$'s that ...
1
vote
1answer
42 views

A very trivial ring homomorphism question!

Ok this question maybe very trivial for all the math geniuses here, but as a beginner, it is not that trivial to me. So, here it goes... Consider the rings $\mathbb{Z}_{3} = \{[0],[1],[2]\}$ and ...
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vote
1answer
34 views

How can the rng $(\mathbb{N}, +, \cdot)$ be an ideal of some ring?

It is known than rng is an ideal of some ring. But how can rng $(\mathbb{N}, +, \cdot)$ be an ideal to some ring? As ring has an inverse element, the first ring we get from $\mathbb{N}$ is ...
3
votes
1answer
29 views

Prove that $R/I$ is free if and only if $I=0.$

Prove that $R/I$ is free if and only if $I=0.$ Is my following proof OK? Assume $R/I$ is a free $R$-module. Then $\mathrm{Ann}(R/I)=0.$ That means $\{x \in ...
1
vote
1answer
45 views

Polynomial Function and Polynomials.

I've got a doubt about a ring of polynomial functions. The problem starting doing this exercise of Fraleigh (The 30). Here I had to show that $P_F$ isn't necessarily isomorphic to $F[x]$. It's easy, ...
1
vote
1answer
9 views

Socle of a ring $R$

It is well-known that for an idempotent $e\in R$, the right $R$-module $eR$ is simple faithful if and only if $Re$ is a simple faithful left $R$-module. Now, I want to prove that when $Re$ is ...
0
votes
2answers
47 views

Every subgroup of $(\mathbb {Z_n},+)$ is closed under multiplication

I am stuck in this proof that every subgroup of $(\mathbb {Z_n},+)$ is also a subring. which requires me to prove it is closed under multiplication. I have to show if $a,b \in G<\mathbb {Z_n}$ , ...
2
votes
2answers
47 views

When does the cancellation law hold for the ring?

Let $R$ be an arbitrary ring. Now, we assume that we don't know whether $R$ has the multiplicative identity or not. I know that $R$ has no zero divisors if and only if the cancellation law holds. So, ...
3
votes
1answer
77 views

Prove that $R[\sqrt{\pi}]$ is a DVR

If $R$ be a DVR(discrete valuation ring) with uniformizer $\pi$, then prove that $R[\sqrt{\pi}]$ is a DVR. How shall I begin, first do I have to find a candidate for the uniformizing element of ...
1
vote
1answer
70 views

Two modules not isomorphic if different number of elements?

Quick question: With $R=\mathbb{Z}, M=\mathbb{Z}, P=2\mathbb{Z}, Q=3\mathbb{Z}$ modules, can I conclude that $M/P = \mathbb{Z}_2 \ncong M/Q = \mathbb{Z}_3$ because they have a different number of ...
0
votes
1answer
26 views

Subfields of $F_2[x] / (x^3 + x + 1)$

What are the subfields of $F_2[x] / (x^3 + x + 1)$? I know $F_2$ is a subfield, and so is itself, but I'm not sure if there are any more. Thanks!
1
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1answer
25 views

Elements of the field $F_2[x] / (x^3 + x + 1)$

What do elements of the field $F_2[x] / (x^3 + x + 1)$ look like? I know this is isomorphic to $F_8$, and that its elements have max degree of 2, so that leaves me with $0$, $1$, $x$, $x^2$, $x+1$ , ...
0
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0answers
21 views

Are there variations on definition of locally finite category?

Wikipedia, under Categorical Algebra, defines a category as locally finite if, for each morphism $m$, the number of factorizations $m = \prod^n m_i$ (with no $m_i$ an identity) is finite. So for ...
1
vote
2answers
34 views

Ideal $\Leftrightarrow$ ~ is a congruence, for $I \subseteq R$ we define $\sim_{I}$ on $R$: $a,b \in R, a \sim_{I} b$ means $a-b \in I$.

Ideal $\Leftrightarrow$ ~ is a congruence. for $I \subseteq R$ we define $\sim_{I}$ on $R$: $a,b \in R, a \sim_{I} b$ means $a-b \in I$. "$\Rightarrow$" Suppose $I$ is an ideal on $R$ which implies ...