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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commutative ring have a unique maximal ideal

Let $R$ is a commutative ring have identity element, and $R$ have a unique maximal ideal. Let $a,b \in R$ such that $\langle a \rangle = \langle b \rangle$. Show that $$a=bu$$ for some $u\in R^*$.
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2answers
323 views

Prove that R is a ring under 'special' definitions of multiplication and addition

Question: Let R be a ring with a 1. Define $\bar R$ to have the same elements of R with addition $$\oplus: a \oplus b = a +b +1$$ andmultiplication $$\otimes: a \otimes b = ab + a +b$$ Prove that ...
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1answer
184 views

$F[x_1,x_2]$ is not a Principal Ideal Domain

If $R$ is a ring (Integral Domain) then $R[x_1,x_2]$ is not a Principal Ideal Domain. Is it a unique factorization domain? Any hints on how to prove its not a Principal Ideal Domain?
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1answer
70 views

Polynomials Question

Let $R$ be a commutative ring with unity, $a\in R$, $f(x) \in R[x]$. Then $a$ is a zero of $f$ iff $x-a$ is a factor of $f$. Solution: If $a$ is a zero of $f$, then by division algorithm, we can ...
4
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1answer
76 views

one simple question in ring theory

Suppose $R$ is a ring (possibly noncommutative), $I$ is a minimal left ideal in it, and $I^2\neq 0$, show that $I=Re$ for some idemopotent $e$. It is easy to show that we can find some $x\in I$, ...
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1answer
384 views

Ring polynomials and Zero Divisors

Let $R[x]$ be a polynomial ring. Show that if $R$ is finite and has zero divisors, $R[x]$ has an infinite number of zero divisors. I'm having trouble wrapping my head around what exactly polynomial ...
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1answer
91 views

Show that $Y^2+Y+1$ is irreducible over the field $\mathbb{Q}[X]/(X^3-2)$

I claimed that $\mathbb{Q}[X]/(X^3-2)$ is a field because $(X^3-2)$ is a maximal ideal of $\mathbb{Q}[X]$.
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2answers
391 views

$R[x]$ has a subring isomorphic to $R$.

Say $R$ is a commutative ring. Does there exists a subring of $R[x]$ that is isomorphic to $R$? My approach would be to define the subring of $R[x]$ that generates $R$. Any thoughts?
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1answer
151 views

Problem related polynomial ring over finite field of intergers

if $f(x)$ is in $F[x]$. $F$ is field of integer mod $p$. $p$ is prime and $f(x)$ is irreducible over $F$ of degree $n$ . prove that $F[x]/(f(x))$ is a field with $p^n$ elements.
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2answers
119 views

What are some function spaces that are UFDs?

What are some "useful" sets of functions (rings under pointwise multiplication) $\mathbb{R} \rightarrow \mathbb{R}$ that are unique factorization domains, other than the polynomial ring? I will ...
1
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1answer
173 views

Field Homomorphisms

Suppose $F$ is a field and $R$ is a ring. The function $f\colon F\to R$ is a surjective homomorphism. Prove that $R$ is either the trivial ring, or $R$ is isomorphic to $F$.
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1answer
91 views

If $A \hookleftarrow B \to R$ each contain $R$, is $R\to A\otimes_B R$ injective?

In this question, all rings and algebras are commutative with identity. Let $R$ be a ring, and let $A$ be an $R$-algebra with an $R$-subalgebra $B$. Suppose that we have an $R$-algebra homomorphism ...
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2answers
119 views

Easy ring questions

Let R be a ring and $p$ a fixed prime number. Then $I_p = \{r \in R $ : additive order of $r$ is a power of $p$ $\}$ is an ideal of $R$. Approach: Pick $r_1,r_2 \in I_p$ and $r \in R$. Then, $a^pr_1 ...
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2answers
305 views

$\langle x,2 \rangle$ is a maximal ideal in $\mathbb{Z}[x]$

Let $\mathbb{Z}[x]/\langle x,2 \rangle$. We know $\langle x,2 \rangle = \{f \in \mathbb{Z}[x] : f(0)$ is even integer$\}$ Also $$\mathbb{Z}[x]/\langle x,2 \rangle = \{f(x) + \langle x,2 \rangle\} = ...
3
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2answers
189 views

If a commutative ring is semiprime and its prime ideals are maximal then it is von Neumann regular (absolutely flat).

If a commutative ring is semiprime and its prime ideals are maximal then it is von Neumann regular (absolutely flat). The converse, although not immediately apparent, can be proven quite easily. But ...
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1answer
127 views

A Gorenstein domain that is not a complete intersection

Could you give me an example (with proof) of a Gorenstein domain that is not a complete intersection?
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1answer
205 views

Help with nilradicals

I know the definition of the nilradical of a ring, and I know that it is an ideal, but I don't know how to "[...] determine the nilradicals of the rings $\mathbb{Z}/(12)$, $\mathbb{Z}/(n)$, and ...
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2answers
171 views

why the addition operation of a ring need to be commutative?

The definition of a ring requires the addition operation to be commutative. But why it has to be?
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1answer
78 views

Why the depth of this modules is 1?

I'm studying on these notes, my question in about a proof on page 63. Basically this is my question: Suppose $R$ local noetherian of positive depth, $M$ a module of finite gorenstein dimension ...
4
votes
5answers
216 views

Primes of the form $a^2+b^2$ : a technical point.

One can classify the prime integers $p$ which can be written as $p=a^2+b^2$ for some integers $a,b\in\mathbb{Z}$ by studying how $p$ decomposes in the ring of Gauss integers $\mathbb{Z}[i]$. Most ...
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1answer
117 views

Subrings of Intergers mod 8

I have to find all the subrings of $Z_8$. I've already deduced that since all subgrings are on subgroups of the additive group, that each must have zero in it, as it acts as the identity element in ...
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3answers
123 views

$(\mathbb{Z}/2\mathbb{Z})[x] /\langle x^2 -2\rangle \ncong (\mathbb{Z}/2\mathbb{Z})[x] /\langle x^2 -3\rangle$

today I have a problem. Let $R_1=\mathbb{Z}_2[x] /\langle x^2 -2\rangle$ and $R_2=\mathbb{Z}_2[x] /\langle x^2 -3\rangle$ prove or disprove $R_1$ and $R_2$ are isomorphic. I felt confuse because ...
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1answer
148 views

Projective Modules over the Ring of Trigonometric Functions

Let $ R = \mathbb{R}[ \cos x, \sin x] $ and consider the ideal $ \langle 1 - \cos x, \sin x\rangle $. Is this ideal a projective module over $R$ ?
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1answer
153 views

Is $\mathbb{Z} \left[ \frac{1+i \sqrt{7}}{2} \right]$ euclidean?

Is the ring $\displaystyle A=\mathbb{Z} \left[ \frac{1+i \sqrt{7}}{2} \right]$ euclidean? If $N : z \mapsto z \overline{z}$, then for all $z \in \mathbb{C}$ there exists $a \in A$ such that ...
2
votes
1answer
86 views

When is a ring considered as a module over a Noetherian subring itself Noetherian?

If $S$ is a ring and $R$ is a Noetherian subring of $S$, and we know $_RS$ (i.e. $S$ viewed as a left $R$-module) is finitely generated (hence Noetherian), is $_SS$ necessarily Noetherian? I can't ...
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1answer
136 views

Isomorphism of cyclic rings

Let $C(l)$ be a cyclic ring with elements $0,e,2e,...,(n-1)e$. Addition is defined by adding coefficients. Multiplication is defined such that $e^2=le$. So if we multiply two elements together, say ...
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2answers
170 views

Verifying a subring of R?

How do I verify that $\mathbb{Z}[\sqrt{2}] = \{ a +b\sqrt{2} \, | \, a,b \in \mathbb{Z} \}$ is a subring of $\mathbb{R}$ ? I'm thinking that i have to show that it's a subgroup which is closed under ...
0
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1answer
70 views

Graded rings and their localizations

Let $A$ be a $\mathbb{Z}_{\geq 0}$-graded ring, $f \in A$ - homogenious, and $I \subset A$ - homogenious ideal. Let $A_f$ be its localization, and $A_{(f)}$ - subring of elements of degree 0. How to ...
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2answers
109 views

Relation between torsion subgroup of multiplicative group of field and solvability of polynomials

In a broad sense, what relationships are there between the torsion subgroup $G$ of the multiplicative group of non-zero elements of a field $K$ and whether or not certain polynomials in $K[x]$ have ...
5
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1answer
90 views

The Auslander dual commutes with ring extensions

Suppose $R$ noetherian and $M$ a finitely generated $R$-module. If you have a projective presentation of $M$: $P_1\rightarrow P_0\rightarrow M\rightarrow 0$, then by dualizing you obtain the following ...
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0answers
70 views

Modules with no regular elements in his annihilator

Suppose $R$ is noetherian, suppose $M$ is a finitely generated $R$-module of grade 0. This means that there are no regular elements in his annihilator. Does this implies that ...
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2answers
244 views

Characteristic of a Quotient Ring

Suppose that $R$ is a commutative ring (with unity) of characteristic $0$, and that the set $Z$ of zero divisors in $R$ forms an ideal. Does it follow that the characteristic of $R/Z$ is $0$? ...
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2answers
67 views

Characteristic of $\Bbb Z_m \times \Bbb Z_n$

I think I have an idea for this, but I'm really struggling on how to put this proof into words. Its clearly the $\text{lcm}(n,m)$. I'm just struggling to communicate why.
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1answer
150 views

Subring of a Field with characteristic P, a prime number.

Suppose $F$ is a field with characteristic $p$, a prime number. Prove that $F$ contains a subring identical to $Z_p$. Are identical subrings the same? There is no mention of this in the text. So do ...
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3answers
305 views

Prove that $(n\mathbb Z, +, \times )$ are the only subrings of $(\mathbb Z, +, \times)$

I had to find all the subrings of the integers and then prove that there aren't any more. It's clear to me the $(n\mathbb Z, +, \times )$ is a subring of the integers for all $n$ element of the ...
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1answer
77 views

Ideal in $\mathbb{Z}[x]$

Is the set of polynomials $a_0+a_1x+\ldots+a_nx^n$, where $2^{k+1}$ divides $a_k$, an ideal in $\mathbb{Z}[x]$? How should I think about this?
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1answer
185 views

On the existence of finitely generated injective modules (Bruns and Herzog, Exercise 3.1.23)

Suppose that $R$ is a local Noetherian ring. Suppose that there exists a non-zero finitely generated injective module $M$. How can I prove that $R$ is Artinian? It is easy if $R$ is ...
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1answer
66 views

When is $\langle m+n{\Bbb Z}\rangle$ a radical ideal in ${\Bbb Z}_n$?

The question is as indicated in the title: When is $\langle m+n{\Bbb Z}\rangle$ a radical ideal in ${\Bbb Z}_n$, i.e. $Rad(\langle m+n{\Bbb Z}\rangle)=\langle m+n{\Bbb Z}\rangle$? I gathered the ...
2
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2answers
138 views

When do we have $Rad(I)=I$ for an ideal $I$ of a ring $R$?

This is kind of a follow-up question about calculating the radical of an ideal. Since $Rad(I)$ is the intersection of all the prime ideals of $R$ that contain $I$, which is a property I learned ...
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4answers
83 views

Is the ring $(\mathbb Z/n, +, \times)$ a subring of $(\mathbb Z, +, \times)$?

Is the ring $(\mathbb Z/n, +, \times)$ a subring of $(\mathbb Z, +, \times)$? I don't think it is, but I can't exactly see why.
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1answer
112 views

Infinitude of irreducibles in subring of an integer ring.

Let $\alpha \in \mathbb{C}$ be an algebraic integer of degree $n$, not a unit, and let $R = \mathbb{Z}[\alpha]$. Then every element $\beta \in R$ can be written uniquely in the form $$c_0+ c_1 ...
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1answer
213 views

How can I calculate the radical of an ideal in ring ${\Bbb Z}_n$?

I learned the concept radical of an ideal from this wikipedia article. I tried some examples and I found that it's not easy to find $Rad(I)$. (That article gives some examples when $R={\Bbb Z}$.) For ...
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1answer
164 views

Discrete Valuation Rings

Let $V = \mathbb A^1(k)$ ($k$ is an algebraically closed field), $\Gamma(V) = k[X]$ and let $K = k(V) = k(X)$. Prove that for each $a \in k = V$, $\mathcal{O}_a(V) := \{f\in K(V): f$ is defined at ...
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0answers
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Is it true that when $ I$ has a term ordering on $R$ where $I$ is an ideal of $R$ then $I$ has a Grobner basis with respect to $\leq$

I don't think that this statement is true. Take for example, $I = (x^2 + y, x^2 y + 1)$. Clearly $I$ $\subseteq$ $k[x,y]$ and can have a term ordering yet $I$ is not a Grobner basis and since it's not ...
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0answers
241 views

On the Nakayama functor

Let $A$ be a finite dimensional $k$-algebra with 1. Denote by $_AP$ the category of projective left $A$-modules finite dimensional. And with $_AI$ the category of injective left $A$-modules finite ...
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1answer
161 views

One to one correspondence of ideals in $R$ and $S^{-1}R$?

I proved the following statement, but I am very unsure that it is correct, since this proposition is not stated in my books for general ideals but only for prime ideals. Please point out where the ...
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1answer
147 views

Categorical definition of the characteristic of a ring

The characteristic of a ring is an important algebraic concept (and a specific number), but it refers to elements, so - in my understanding - it is evil (from the point of view of category theory). So ...
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1answer
60 views

Contraction of the ideal $S^{-1}Q$

The following is a statement about extended and contracted ideals under localisation from Atiyah, McDonald They use that for any ideal I considering the homomorphism $f: A \rightarrow S^{-1}A$ ...
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0answers
37 views

Does an irreducible polynomial in K(t)[x] give an irreducible polynomial in K[t][x]

Let $K[t]$ be the ring of polynomials over a field $K$. Let $K(t)$ be its fraction field. Let $f$ be an irreducible polynomial in $K(t)[x]$. There exists an element $a\in K[t] $ such that $af$ is in ...
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1answer
85 views

Levels of Rings and Fields, -1 as a sum of squares

Definition: Let $R$ be a commutative ring. The level of $R$, denoted $s(R)$, is the least positive integer $s$ such that $-1$ can be written as the sum of $s$ many squares in $R$. Set $s(R)=\infty$ if ...