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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Every right principal ideal non-emptily intersects the center — what is that?

This is a follow-up to Do Lipschitz/Hurwitz quaternions satisfy the Ore condition? Jyrki Lahtonen answered the question in the positive by noticing that every right principal ideal in either ring has ...
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2answers
66 views

Do Lipschitz/Hurwitz quaternions satisfy the Ore condition?

The Lipschitz quaternions $L$ are the quaternions with integer coefficients and the Hurwitz quaternions $H$ are the quaternions with coefficients from $\Bbb Z\cup(\Bbb Z+1/2).$ A ring satisfies the ...
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3answers
23 views

Ideal generated by a subset of ring.

The definition of Ideals generated by a subset : Let $S$ be any subset of ring $R$ then an ideal $I$ of $R$ is said to be generated by $S$ if : (1) $S \subseteq I$. (2) for any ideal $J$ of $R$ ...
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3answers
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The subring criterion

As we know that the subring criterion states that a subset $H$ of ring $R$ is a subring if and only if : (1) $H$ is non-void , and (2) for all $x,y \in H$,$x-y \in H$. (3) product $xy \in H$ . The ...
2
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1answer
44 views

1) Show that $(\Bbb Z[\sqrt2]^*, .)$ is infinite.

1) Show that $(\Bbb Z[\sqrt2]^*, .)$ is infinite. 2) Classify $(\Bbb Z[\sqrt2]^*, .)$, where $\Bbb Z[\sqrt2]^*$ is the group of units of $\Bbb Z[\sqrt2]$ What I have done so far that for $a+b\sqrt2$ ...
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1answer
31 views

$R$ is a commutative integral ring, $R[X]$ is a principal ideal domain imply $R$ is a field

I've just read a proof of the statement: Let $R$ be a commutative integral ring. If $R[x]$ is a principal ideal domain, then $R$ is a field. In one part of the proof there is a step which I don't ...
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2answers
67 views

Why $R[X]$ is never a field? [on hold]

If $R$ is a ring, why $R[X]$ is never a field ? I hope the question is not to ambigus but it's exactly the assignment.
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1answer
36 views

$\mathbb{Q}[x,y]/(f(x,y))$ is not a field for any irreducible $f(x,y) \in \mathbb{Q}[x,y]$.

I need to show that $\mathbb{Q}[x,y]/(f(x,y))$ is not a field for any irreducible $f(x,y) \in \mathbb{Q}[x,y]$. My approach was to find a bigger proper ideal containing $f(x,y)$ but i am unable to ...
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0answers
17 views

Factor rings of polynomial rings

Is there a unified explanation to the following phenomena? 1) $\mathbb{R} [X, Y] / (X^2 + Y^2 - 1) \mathbb{R} [X, Y]$ is not a UFD. 2) $\mathbb{C} [X, Y] / (X^2 + Y^2 - 1) \mathbb{C} [X, Y]$ is a ...
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1answer
42 views

Sets of prime ideal contain a minimal element

I want to prove that every nonempty set of prime ideal contain a minimal element, my attempt is to prove it by using zorns lemma and i would like to know if my proof is valid. Let $\Sigma$ be a ...
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3answers
156 views

Can we say “commutative ring = field”?

We know the difference between ring (R) and field (F) is that R does not guarantee multiplication is commutative. Now, if considering commutative R, which means (R,.) is a group, can we say: ...
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1answer
34 views

Colon ideal of fractional ideals is itself a fractional ideal

I received this question on homework in my homological algebra class and I need some guidance. Let $R$ be a commutative integral domain and $K$ be its field of fractions. A fractional ideal $I$ of ...
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1answer
24 views

Ring Theory: Showing sets are subrings

Let S=C[0,1] be the set of real-valued continuous functions defined on the closed interval [0,1], where we define f+g and fg, as usual, by (f+g)(x)=f(x)+g(x) and (fg)(x)=f(x)g(x). Let 0 and 1 be the ...
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1answer
56 views

How to show that $\mathbb{C}[x_1,x_2,x_3, x_4]/(x_1x_2 - x_4x_3, x_1x_3 - x_2x_4, x_4x_1 - x_3x_2)$ is integral domain

I am looking for a way to show that the ring $\mathbb{C}[x_1,x_2,x_3, x_4]/I$ where $I = (x_1x_2 - x_4x_3, x_1x_3 - x_2x_4, x_4x_1 - x_3x_2)$ is an integral domain. In other words I want to show $I$ ...
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2answers
48 views

identify the ring $\mathbb{Z}[x]/(2x-1)$

Suppose it asks to show $\mathbb{Z}[x]/(2x-1) \cong \mathbb{Z}[\frac{1}{2}]$ Cand I do like this ? First of all elements of $\mathbb{Z}[x]/(2x-1)$ is of form $\frac{m}{2^n}+(2x-1)$ and also all ...
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2answers
32 views

Prove that, if $(a)=(a')$, then $a'=ua$

Let $R$ be integral domain. Show that if $2$ principal ideals $(a)$ and $(a')$ are equal (where $a,a'\in R$) then there exists $u\in R^{\times}$ such that, $a'=ua$ Now if $(a)=(a')$ then ...
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3answers
95 views

Every non-unit is in some maximal ideal

I am trying to prove that every non-unit of a ring is contained in some maximal ideal. I have reasoned as follows: let $a$ be a non-unit and $M$ a maximal ideal. If $a$ is not contained in any maximal ...
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23 views

Exercise related to commutative ring and finitely generated ideals

Let $R$ be a commutative ring with $1 \neq 0$. An ideal $I$ of $A$ is finitely generated if there are $r_1,...,r_n \in R$ such that $I=<r_1,...,r_n>$. Let $S$ be a multiplicative set of $R$. ...
3
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0answers
29 views

Non Maximal Prime ideal! [duplicate]

Assume $S$ to be all continuous functions from $[0,1]$ to $\mathbb R$. I know by compactness of $[0,1]$ it follows that all maximal ideals of $S$ have the form $M_{x_0}=\{f\in S \mid f(x_0)=0\}$.Does ...
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2answers
56 views

Commutative ring and maximal ideal problem

Let $A$ be a commutative ring and $M$ be a proper maximal ideal in $A$. Show the following properties: (a) If each $a \in A \setminus M$ is a unit element in $A$, then $M$ is the only maximal ideal ...
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1answer
36 views

Why is axiom of choice needed? (Equivalent conditions for Noetherian)

Let $R$ be a commutative ring with $1$ and let $M$ be a left $R$-module. On page 458 of Dummit and Foote's Algebra, 3rd edition, they show that $M$ is Noetherian (i.e. satisfies A.C.C. on submodules) ...
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0answers
52 views

Find the units of the ring $\mathbb{Z}_6[x]/\langle 2x+4 \rangle$. [on hold]

Find the units of the ring $\mathbb{Z}_6[x]/\langle 2x+4 \rangle$.
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1answer
29 views

Proving a submodule is Noetherian

I'm doing some independent study with a professor in Ring/Field theory (I'm an undergraduate) and I have been having a hell of a time wrapping my head around problems involving Noetherian rings and ...
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1answer
48 views

A property of minimal prime ideal

Let $R$ commutative ring with unity, $S\subseteq R$ subring, $p$ minimal prime ideal of $S$. Show there exists a minimal prime ideal $q$ in $R$ with the property that the contraction $q^c=q\cap S=p$. ...
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1answer
23 views

How to prove that the evaluation map is a ring homomorphism?

This is a really easy question, but I'm stuck in the logic of it... Let $F$ be an integral domain and $F[x]$ its polynomial ring. Let $a\in F$ fixed, define $\phi: F[x]\to F$ as ...
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0answers
21 views

Gröbner basis, Buchberger algorithm, ideal and ring [closed]

I major in electrical engineering and for my thesis I'm dealing with a paper about three view triangulation http://link.springer.com/chapter/10.1007%2F978-3-540-76390-1_54#page-1 I'm not familiar with ...
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0answers
47 views

Necessary and sufficient condition for a ring homomorphisms property

The question states: Let $R$ be a commutative ring with unity and let $A,B\subseteq R$ be two ideals, find a necessary and sufficient condition for $\mathrm{Hom}(R/A,R/B)=0$. Since ...
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1answer
53 views

Find a generator for $(f,g)$, two polynomials in $\mathbb Q[x]$

I have two polynomials $$ \def\f{x^5+2x^4+3x^3+3x^2+2x+1} \def\g{x^5+3x^4+4x^3+4x^2+2x+1} \def\s{\{r f + s g : r,s\in\mathbb Q[x]\}} \def\gcd{x^2+x+1} f=\f\\ g=\g $$ I want find a polynomial that ...
2
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1answer
66 views

Self-injective ring but not semisimple?

It is well-known that if $K$ is a field, then $K[x]/(f(x))$ is a self-injective ring for any polynomial $f(x)$ in $K[x]$. On the other hand, we know that a ring $R$ being semisimple is equivalent to ...
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1answer
45 views

Preimage of maximal ideal is maximal [duplicate]

I've just started a commutative algebra course and I'm stuck on the very first homework problem: Let $A \not= \{0\}$ be a commutative ring. Let $\Phi : A \longrightarrow B$ be a ring homomorphism ...
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0answers
32 views

Finite Division Rings are Fields [duplicate]

I have seen a problem recently. It says that every finite division ring is a field. How to show this?
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1answer
38 views

If $x \in R$ is irreducible then $x u$ and $xy$ are irreducible where $u \in R^*$ and $y$ is irreducible.

If $x \in R$ is irreducible then $x u$ and $xy$ are irreducible where $u \in R^*$ is a unit and $y \in R$ is irreducible. Let $R$ be a ring. How do I see that if $x \in R$ is irreducible then: ...
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2answers
102 views

A question on ring homomorphisms and maximal ideals.

Let $A,B$ be commutative rings, and let $\phi: A \to B$ be a ring homomorphism where $B$ has finitely many elements. Prove that if $I \subset B$ is a maximal ideal then $\phi^{-1}(I)$ is also a ...
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1answer
52 views

kernel of homomorphism $\mathbb{C}[x,y] \to \mathbb{C}[t]$ but in general case

Let $f:\mathbb{C}[x,y] \to \mathbb{C}[t]$ be a homomorphism that is identity on $\mathbb{C}$ and sends $x\to x(t),y \to y(t)$ and such that $x(t),y(t)$ aren't both constant. Prove $ker(f)$ is a ...
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1answer
96 views

$A = B\cdot p(A)$. Show $A$ and $B$ commute.

A problem my professor sent out: Suppose $p$ is a polynomial with constant term nonzero. Suppose $A,B\in M_n(\mathbb{C})$ such that $A=B\cdot p(A)$. Show that $A$ and $B$ commute. This is a ...
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0answers
64 views

If every element of a ring is either potent or central, the ring is commutative

Let $R$ be a ring such that every element is potent ($x^k = x$, for some integer $k>1$) or central. Prove that $R$ is commutative. My prove: Let $x,y$ be elements of $R$, suppose one of them ...
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1answer
27 views

Rational function in both $k(X)[Y]$ and $k(Y)[X]$

If I have a rational function in $X$ and $Y$ and it can be written as both a polynomial in $Y$ with coefficients being rational functions in $X$ (that is, an element of $k(X)[Y]$) and as a polynomial ...
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46 views

Realize groups as unit group of a ring

Let $A$ be a ring, $G$ be a group, and $f:A^{\times} \rightarrow G$ be a group homomorphism. Is there any ring $B$ and ring homomorphism $\varphi:A \rightarrow B$ such that $G$ is subgroup of ...
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1answer
28 views

Ring Theory (idempotents)

Let S=C[0,1] be the set of real-valued continuous functions defined on the closed interval [0,1], where we define f+g and fg, as usual, by (f+g)(x)=f(x)+g(x) and (fg)(x)=f(x)g(x). Let 0 and 1 be the ...
0
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1answer
28 views

Let $I=(2,X)$ and $J=(3,X)$ be ideals of $\mathbb{Z}[X]$. Prove $V=\{i \cdot j : i \in I, j \in J\}$ is not an ideal. [closed]

Let $I=(2,X)$ and $J=(3,X)$ be ideals of $\mathbb{Z}[X]$. Prove $V=\{i \cdot j : i \in I, j \in J\}$ is not an ideal. Possible strategy: find $v_1,v_2 \in V$ such that $v_1+v_2 \notin V$.
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3answers
52 views

Zero divisors and units of $\mathbb Z[X]/ \langle X^3 \rangle$

Problem: Find the zero divisors and the units of the quotient ring $\mathbb Z[X]/\langle X^3 \rangle$. If $a \in \mathbb Z[X]/ \langle X^3 \rangle$ is a zero divisor, then there is $b \neq 0_I$ ...
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1answer
49 views

$\mathbb R[X] /<X^2-1>$ and $\mathbb R[X,Y]/<XY>$ are not fields

I have to prove that 1)$\mathbb R[X] /<X^2-1>$, and 2) $\mathbb R[X,Y]/<XY>$ are not fields. So, I must exhibit an element $r$ from say $\mathbb R[X] /<X^2-1>$ that has no ...
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0answers
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prove that $(E_{p^n},*)$ is cyclic group

if $p \in$ $\mathbb{N}$ is a prime integer, how can i prove that $E_{p^n}$ the group of invertible elements of $\frac{\mathbb{Z}}{p^n\mathbb{Z}}$ is a cyclic group.
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3answers
74 views

Set containing all rings!

Does there exist a set containing all rings ? Possible idea :I think such set is not possible.If S is a set containing all rings i think we can again define a structure on S to make it Ring and that ...
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1answer
36 views

Ring homomorphism $f: \mathbb C \to \mathbb C$ such that $f(\mathbb R) \subset \mathbb R$

Exercise Find all the ring homomorphisms $f: \mathbb C \to \mathbb C$ such that $f(\mathbb R) \subset \mathbb R$ My attempt at a solution In a previous problem I've already proved that the only ...
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1answer
18 views

Statements about ring homomorphisms and division rings

Problem Decide whether the following statements are false or not. 1) If $A$ is a commutative ring such that every ring homomorphism different from the null morphism $\phi:A \to A'$ is injective, ...
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41 views

An error in the book “noncommutative ring” writed by Herstein

I'm reading the book "noncommutative ring" writed by Herstein. In the page 15, the author says that Let $F$ be a field and $A$ is an algebra over $F$. Let $\rho$ be a maximal regular right ideal ...
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1answer
16 views

End(V) and End(V)xEnd(V) are isomorphic

Let R=End(V) be the ring of all linear endomorphisms of an infinite dimension complex vector space V with countable basis $\{e_{1},e_{2},...\}$ . Prove that R and RxR are isomorphic as left R-modules. ...
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1answer
30 views

How do I take the contraction of an ideal which is not in the image of the given morphism?

If I have a morphism of rings $\phi: A \to B$ which is not surjective, how should I take the preimage of an ideal not contained in the image of $\phi$?
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2answers
53 views

Let $a$ and $b$ be two elements in a commutative ring $R$ and $(a, b) = R$, show that $(a^m, b^n) = R$ for any positive integers $m$ and $n$.

I stumbled across a question that I have no idea how to start. I know the questions asking to show that the multiples of $a$ and $b$ as an ordered pair make still make the whole ring. Any sort of ...