This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

learn more… | top users | synonyms (2)

0
votes
2answers
25 views

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
votes
1answer
67 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 $a+b\sqrt2$ ...
2
votes
2answers
287 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 ...
1
vote
1answer
49 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 ...
1
vote
0answers
36 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 ...
3
votes
1answer
115 views

Set of prime ideals contain a minimal element

I want to prove that every nonempty set of prime ideals contain a minimal element. My attempt is to prove it by using Zorn's lemma and i would like to know if my proof is valid. Let $\Sigma$ be ...
2
votes
3answers
242 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 ...
1
vote
1answer
140 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 ...
1
vote
1answer
39 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 ...
6
votes
1answer
100 views

How to show that $\mathbb{C}[x_1,x_2,x_3, x_4]/(x_1x_2 - x_3x_4, x_1x_3 - x_2x_4, x_1x_4 - x_2x_3)$ is an 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 $$I = (x_1x_2 - x_3x_4, x_1x_3 - x_2x_4, x_1x_4 - x_2x_3)$$ is an integral domain. In other words I want to show ...
1
vote
2answers
117 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 ...
2
votes
2answers
53 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 ...
3
votes
3answers
581 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 ...
3
votes
0answers
46 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
85 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 ...
2
votes
1answer
123 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
53 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 ...
2
votes
1answer
133 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 $q^c=q\cap S=p$. ...
2
votes
1answer
396 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 ...
2
votes
0answers
84 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
70 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
110 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 ...
4
votes
1answer
378 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
votes
0answers
38 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
vote
1answer
40 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: ...
3
votes
2answers
251 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 ...
2
votes
1answer
92 views

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

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
106 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 ...
1
vote
0answers
110 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
36 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 ...
2
votes
1answer
75 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 $0$ ...
2
votes
3answers
99 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 $b \neq 0_I$ ...
2
votes
1answer
75 views

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

I have to prove that 1) $\mathbb R[X] /\langle X^2-1\rangle$ is not a field, and 2) $\mathbb R[X,Y]/\langle XY\rangle$ is not a field. So, I must exhibit an element $r$ from say $\mathbb R[X] ...
2
votes
0answers
29 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.
1
vote
3answers
101 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
49 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
36 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, ...
0
votes
0answers
60 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 ...
1
vote
1answer
45 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
36 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
72 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
47 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
120 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
125 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 $R$?
2
votes
1answer
194 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 $V$. ...
8
votes
1answer
197 views

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
2answers
177 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
78 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
31 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
53 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 ...