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
20 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 ...
1
vote
2answers
49 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 ...
1
vote
1answer
28 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) ...
0
votes
1answer
42 views

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

Find the units of the ring $\mathbb{Z}_6[x]/\langle 2x+4 \rangle$.
0
votes
1answer
25 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 ...
1
vote
1answer
41 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 $q^c=q\cap S=p$. ...
1
vote
1answer
19 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 ...
0
votes
0answers
21 views

Gröbner basis, Buchberger algorithm, ideal and ring [on hold]

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 ...
2
votes
0answers
43 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 ...
2
votes
1answer
51 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
votes
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 ...
2
votes
1answer
41 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 ...
0
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0answers
31 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?
1
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1answer
37 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: ...
2
votes
2answers
88 views

A question on ring homomorphisms and maximal ideals.

Let $A,B$ be commutatximal ideal in $A$
2
votes
0answers
39 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 ...
6
votes
1answer
94 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 ...
0
votes
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 ...
0
votes
1answer
26 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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votes
0answers
45 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 ...
1
vote
1answer
24 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
votes
1answer
26 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. [on hold]

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 $v_1+v_2 \notin V$.
2
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3answers
50 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 $b \neq 0_I$ ...
1
vote
1answer
47 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 ...
2
votes
0answers
26 views

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.
0
votes
3answers
67 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 ...
2
votes
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 ...
0
votes
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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votes
0answers
40 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 ...
0
votes
1answer
14 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. ...
0
votes
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 $\phi$?
2
votes
2answers
51 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 ...
1
vote
1answer
38 views

Ideal and factor ring

I want to determine the ideals and factor rings for $R\times\mathbb{Z}_{116}$ and $Q \times\mathbb{Z}_9$ I know that $Q$ and $R$ are fields and their ideals are ${0}$ and $Q / R$ and the ideals in ...
1
vote
1answer
48 views

Find all the ideals of $\mathbb Q[X]$

I am trying to find all the ideals of the ring $\mathbb Q[X]$. If $I$ is a non trivial ideal of $\mathbb Q[X]$, then there exists $p(x) \in \mathbb Q[X]$. Since $I$ is an ideal and a group under ...
2
votes
2answers
70 views

Are ideals necessarily definable?

Consider the first-order language of rings. Let $R$ be a ring and $I \subseteq R$ be an ideal. Is $I$ necessarily $\emptyset$-definable? If not, what if we allow parameters from $R$?
1
vote
1answer
32 views

Left ideals of $M_n(K)$ [duplicate]

Let $K$ be a field and $n \in \mathbb N$. Show the following (i) Let $V \subset K^n$ be a subspace and $I_V$ the subset of $M_n(K)$ consisting of all the matrices whose rows belong to $V$. Prove that ...
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0answers
97 views
+200

Recovering free modules from their projective limit

Let $\dotsc A_2 \to A_1 \to A_0$ be a sequence of surjective homomorphisms of commutative rings. Consider the projective limit $\varprojlim_i A_i$. If $S$ is an (infinite) set, then $\varprojlim_i ...
0
votes
0answers
13 views

Reversible and Directly Finite Rings? [closed]

A ring $R$ is reversible if $ab=0$, then $ba=0$. A ring $R$ is directly finite if $ab=1$, then $ba=1$. Give an example of a non-commutative ring that is reversible? Give an example of a ...
0
votes
2answers
36 views

Units in $\mathbb Z[\sqrt{2}]$ [duplicate]

I am trying to find all units in $\mathbb Z[\sqrt{2}]$. Suppose $x=a+b\sqrt{2}$ is a unit. Then there is $y=c+d\sqrt{2}$ such that $$xy=(a+b\sqrt{2})(c+d\sqrt{2})=1$$ So $$ac+2bd+(ad+bc)\sqrt{2}=1$$ ...
3
votes
2answers
52 views

Zero divisors of $C[0,1]$ [duplicate]

Find the zero divisors of the ring $R=C[0,1]$ the continuous functions $f:[0,1] \to [0,1]$. I could thought of a set $S$ that I think is included in the set of zero divisors, but I am not sure if $S$ ...
0
votes
1answer
24 views

Prove a particular set is a ring with unity

I have to show that $(\mathbb{Z}[G],+,.)$ is a unitary ring, where $$\mathbb Z[G]=\{\sum_{g \in G} a_g.g| a_g \in \mathbb Z, a_g \neq 0, \text{only for finite g in G}\}$$ with $G$ group and $(\sum ...
0
votes
0answers
48 views

Cardinality of ring having more than one left inverse for some element! [duplicate]

Suppose $R$ is a ring with unity $1$ and for some $a\in R$ there exists more than one left inverse of $a$ in $R$. Show that $R$ has infinitely many left inverses of $a$. I am trying to define a ...
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vote
4answers
61 views

Ring Theory question in a GRE practice exam

I have a question about a GRE practice problem relating to Rings. The question is as follows: Suppose that two binary operations, denoted by $\oplus$ and $\odot$ , are defined on a nonempty set $S$, ...
1
vote
1answer
20 views

If a ring has its field of fraction as algebraic number field $K$, would this ring be $O_K$?

Suppose that ring has its field of fraction as algebraic number field $K$. Would this ring then be $O_K$, ring of integers? Also, for $O_K$, would subring of $O_K$ be integrally closed?
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0answers
34 views

Reference about Tensor Product

do you know some reference about tensor product of modules, with all elementary properties are proved ?? I want something a bit more explicit than "Commutative Algebra" of Atiyah and Macdonald. ...
0
votes
1answer
29 views

Center of a ring matrix

I have the following ring matrix: Ep= a b pc pu+v where a,b,c,u,v belongs to Zp where p is a prime now the center of this matrix is: Z(Ep)= ...
1
vote
1answer
40 views

A Commutative Ring Having a Unique Prime Ideal (Dummit and Foote, Prob 7.4.40(i))

I am trying to solve Problem 7.4.40 from Dummit and Foote, a part of which states: Let $R$ be a commutative ring with $1\neq 0$ such that $R$ has exactly one prime ideal. Then every ...
2
votes
2answers
31 views

Linear equation over $\mathbb{Z}/n\mathbb{Z}$

For given $a,b\in \mathbb{Z}/n\mathbb{Z}$ is there a criterion which allows one to determine whether there exists $x\in \mathbb{Z}/n\mathbb{Z}$ with $ax=b$?
0
votes
1answer
27 views

What is wrong in this proof where I show that $\text{End}_k(k^2)$ is a division ring?

I've proven that $R=\text{End}_k(k^2)$ where $k$ is a field, is simple. But out of this, I think that this should imply that $\text{End}_k(k^2)$ is a division ring, which is obviously false, but what ...
0
votes
0answers
27 views

A question about the number of generators of an ideal.

Is the number of generators of an ideal always constant? I can't seem to find an online reference that proves/disproves this fact. It would be easy to prove in linear algebra, but this case is ...