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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-4
votes
1answer
79 views

Commutative domain with two maximal ideals of different heights [closed]

Give an example of a commutative domain $R$ and two maximal ideals $\mathfrak{m}_1, \mathfrak{m}_2$ in $R$ of different heights.
1
vote
1answer
12 views

$IM$ not finitely generated , $J \subseteq I$, $JM$ finitely generated; is there some $a\in I$ such that $JM\subsetneq\langle a,J\rangle M$?

Let $R$ be a commutative ring with unity, $M$ be an $R$-module, $I$ be an ideal of $R$ such that $IM$ is not a finitely generated submodule. Let $J \subseteq I$ be a finitely generated ideal such that ...
0
votes
2answers
57 views

If $0 \rightarrow M' \rightarrow M \xrightarrow[]{f} M'' \rightarrow 0$ is exact then $M$ is Noetherian iff $M',M''$ are. [duplicate]

If $0 \rightarrow M' \rightarrow M \xrightarrow[]{f} M'' \rightarrow 0$ is exact then $M$ is Noetherian iff $M',M''$ are. For an infinite chain $M_i$ in $M$ we have chains $M'_i = M_i \cap M'$ ...
0
votes
3answers
79 views

Why can't $0_R$ have an inverse under multiplication?

A definition of division ring says that its every element has an inverse under multiplication, except $0_R$, where $0_R$ is the additive identity. Why can't $0_R$ have such an inverse too?
0
votes
1answer
26 views

$M$ be a finitely generated $R$-module , and $N$ be a submodule of $M$ ; is it possible to have a meaning for $Ann(M)/N$ as an ideal?

Let $M$ be a finitely generated $R$-module, and $N$ be a submodule of $M$; is it possible to have a meaning for $Ann(M)/N$ as an ideal? (I ask this question due to its use in the third line in the ...
2
votes
0answers
53 views

Definibility of $\mathbb{Z}$ in product rings

If $R$ is a product ring whose factors are in a finite number and are all quotients of $\mathbb{Z}$ (that is, either $\mathbb{Z}$ or $\mathbb{Z}_n$'s ), is it a sufficient and necessary condition for ...
1
vote
0answers
27 views

An Ideal in the Group Ring $RG$

So I'm working on Abstract Algrebra (Dummit & Foote). Let $R$ be a commutative ring with identity $1$ and let $G=\{g_1, ..., g_n\}$ be a finite group. Prove that $$I=\{\sum_{i=1}^n ag_i | a\in ...
1
vote
1answer
26 views

Questions of a completely reducible module

Please help to deal with the tasks of: $1)$ Which cyclic groups are completely reducible as a $\mathbb Z$-modules? $2)$ Which cyclic modules are completely reducible over the ring $\mathbb F[x]$, ...
0
votes
3answers
65 views

Is there any trivial ring which isn't null?

By a trivial ring, I mean one that fulfills the following: ${\forall}x,y\,{\in}\,R:xy=0_R$ A null ring is a ring with only one element. So far I couldn't think of any trivial ring which isn't null. ...
3
votes
1answer
34 views

Jacobson radical of a noncommutative ring

If R is a commutative rings I know that J(R), the Jacobson radical of $R$, coincides to all $r \in R$ such that the elements $$1+s_1rs_2\in U(R)$$ are units of $R$ for any $s_1,s_2 \in R$. My ...
0
votes
1answer
29 views

a cyclic nonsimple module over $\mathbb{Z}$

Let $M$ be a module over $\mathbb{Z}$. Prove that: a.$M$ is a cyclic module if and only if it's a cyclic group. b.$M$ is simple if and only if it's cyclic of prime order. I tried ...
1
vote
2answers
60 views

Show that for a commutative ring $R$, there always exists a ring homomorphism from $R$ to some field.

Show that for a commutative ring $R$, there always exists a ring homomorphism from $R$ to some field. I try to extend $R$ to a division ring by throwing in multiplicative inverse for each ...
1
vote
1answer
53 views

Showing there are no nontrivial ring homomorphisms from $\mathbb{Z}\to\mathbb{Z}$ [duplicate]

I have: If $\phi:\mathbb{Z}\to\mathbb{Z}$ is a homomorphism, then $f(1)=f(1\ast1)=f(1)\cdot f(1)$. Then $$0=f(1)\cdot f(1)-f(1)=f(1)\cdot\left[f(1)-\epsilon\right]$$ implies that $f(1)=0$ or ...
0
votes
2answers
37 views

Find all prime ideals and maximal ideals of $\mathbb{Z}/12\mathbb{Z}$

How do I go about finding these? I know that the prime and maximal ideals in this case are the same, and that an ideal $M$ is only a maximal ideal of $R$ iff $R/M$ is a field, but I don't really know ...
1
vote
1answer
22 views

Let $\phi:R\to R'$ be a ring homomorphism.

Let $\phi:R\to R'$ be a ring homomorphism. Let $N'$ be either an ideal of either $\phi[R]$ or of $R'$. Show that $\phi^{-1}[N']$ is an ideal of $R$. My attempt so far: $N'$ is an ideal of $R'$ or ...
2
votes
1answer
59 views

Commutativity of “extension” and “taking the radical” of ideals

Let $K$ be a field (not necessarily algebraically closed) and $\overline{K}$ its algebraic closure. By $K[\text{X}]$, I mean $K[X_1,...,X_n]$. Is it true that the operations of "extension" and ...
0
votes
1answer
39 views

Is an integrally closed domain of Krull dimension at most $2$ a Krull domain? [closed]

Let $D$ be an integrally closed domain of Krull dimension at most $2$. Is $D$ a Krull domain?
1
vote
1answer
23 views

Ring module homomorphism

Got peculiar question, not sure if my answer is correct. Let $R$ be ring and $b,r\in R$ Let also $f_b:R\rightarrow R$ where $f_b(r)=rb$ and $g_b(r)=br$ Why $f_b$ is R-module homomorpism, but $g_b$ ...
0
votes
3answers
58 views

What do the Squared Brackets stand for in $\mathbb{Z}[\mathrm{i}] $ [duplicate]

What do the Squared Brackets stand for in e.g. $\mathbb{Z}[\mathrm{i}] $?
-4
votes
1answer
47 views

Every nonzero element $x\in\mathbb Z_n$ is either a unit or a zero divisor [closed]

I'd like to show that every nonzero element $x\in\mathbb Z_n$ is either a unit or a zero divisor, i.e. for every $x\in\mathbb Z_n$ there exists either $x'\in\mathbb Z_n$ such that $x'x=1$, or ...
2
votes
1answer
66 views

Direct sums of semisimple objects

Let $\mathcal{A}$ be an abelian category. Call an object $M\in\mathcal{A}$ semisimple if every exact sequence $0\longrightarrow M'\longrightarrow M\longrightarrow M''\longrightarrow 0$ splits. Is it ...
2
votes
2answers
41 views

Compute the (multiplicative) inverse of $4x+3$ in the field $\frac {\Bbb F_{11}[x]}{\langle x^2+1 \rangle}$?

So I am finding a polynomial $px+q$ ($p,q \in \Bbb F_{11}$) which is multiplicative inverse of $4x+3$ in $\frac {\Bbb F_{11}[x]}{\langle x^2+1 \rangle}$. i.e. $[(4x+3)+\langle x^2+1 ...
2
votes
1answer
25 views

Finding a prime in a ring extension using Nakayama's lemma

This is a follow up to my previous question here if $A \subset B$ is a finite ring extension and $P$ is a prime ideal of $A$ show there is a prime ideal $Q$ of $B$ with $Q \cap A = P$. (M. Reid, ...
0
votes
0answers
26 views

Explanation of the term rings [duplicate]

why do we call rings rings ? Is it random name or is it because of some structural property?
3
votes
5answers
51 views

Given ring $F[X]/(X^2)$ why is the ideal (X) the unique maximal ideal of the ring [duplicate]

Given ring $F[X]/(X^2)$ I'm trying to understand why the ideal (X) is the unique maximal ideal of the ring. I have figured out that an element in the ring is either in the ideal (X) or is a unit, but ...
2
votes
2answers
54 views

Showing there is a prime in a ring extension using Nakayama's lemma

Here's the problem that I'm working on: if $A \subset B$ is a finite ring extension and $P$ is a prime ideal of $A$ show there is a prime ideal $Q$ of $B$ with $Q \cap A = P$. (M. Reid, ...
1
vote
1answer
43 views

Prove whether R is integral domain.

I'm having trouble with figuring out whether a given ring is an integral domain or not. This comes from my confusion about the zero element. This ring R is a commutative triple $(Z,*,o)$ with ...
-4
votes
1answer
24 views

Subring of a field [closed]

Let $R$ be a subring of a field $F$ such that for each $x\in F$ either $x\in R$ or $x^{-1}\in R$. Prove that if $I$ and $J$ are two ideals of $R$, then either $I\subseteq J$ or $J\subseteq I$.
1
vote
1answer
50 views

The category of (completable) rings has enough projectives in it

I am working on functors and projective resolutions and of course the issue of "Enough projectives" comes up. I know $R$-modules have enough but I am curious about the category of rings in general? ...
0
votes
0answers
41 views

nil Jacobson radicals

Let $f$ be an idempotent element in a ring $S$ with Jacobson radical $J$ so that both $fJf$ and $(1-f)J(1-f)$ are nil. I guess that $J$ is nil too, but I am not sure. I know that the former is ...
0
votes
1answer
59 views

Can you construct a field of characteristic $\neq 0, 2$ such that every one of its subrings is also a field?

A friend asked me this a few days ago, and I was thinking that it may be impossible, but now I'm not so sure. He suggested a "nonprincipal ultrapower" $(\mathbb{Z}/(2))^{N}$ such that every subring is ...
1
vote
0answers
36 views

Reference for an isomorphism

Let $A$ be a finite dimensional algebra over a field $K$ and $D:=Hom_K(-K)$ the natrual duality of mod-$A$. Let $M$ and $N$ be $A$-bimodules. Then there is an isomorphism $A$-bimodules: $Hom_A(M,D(N)) ...
0
votes
1answer
33 views

Prove that $\text{nil}(\overline{R})=\sqrt{A}/A$

Let $R$ be a commutative ring and $A \lt R$ an ideal. Define the radical of an ideal $A$ to be $\sqrt{A}:=\lbrace x \in R \mid x^n \in A \text{ for some } x \in \mathbb{Z}^+ \rbrace$. Let ${}^{-}:R ...
6
votes
2answers
95 views

Ideal of $\mathbb{C}[x,y]$ not generated by two elements

Consider the ring of polynomials in two variables $\mathbb{C}[x,y]$. Show that the ideal $\langle xy^3, x^2y^2, x^3y\rangle$ cannot be generated by two elements. Until now, I assumed by ...
0
votes
1answer
24 views

the associate of a prime is prime in integral domain

I was hoping someone could give me a hand getting started trying to prove that in an integral domain, if a and b are associates, then a is prime if and only if b is.
-1
votes
0answers
23 views

Prove the Chinese Remainder Theorem for multiplicative groups [duplicate]

Suppose $n=p_1^{k_1}p_2^{k_2} \ldots p_n^{k_n}$ where $p_i$ are distinct primes. Show that $\mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}/p_1^{k_1}\mathbb{Z} \times \ldots ...
1
vote
0answers
53 views

A general definition for a character of a (not necessarily associative) algebra

Let $A$ be a algebra over a algebraically closed field $k$. Is there certain definition of a "character" $f: A \rightarrow k$? That is, what is the common and useful condition for a linear map $f: A ...
1
vote
1answer
36 views

Ring homomorphisms that generate the unit ideal

We generally do geometry over commutative rings by insisting that all rings have a $1$, and that morphisms preserve $1$. This corresponds, roughly, to the following geometric property: if $f:X\to Y$ ...
0
votes
1answer
30 views

Prove that the quotient ring $R/(f)$ has four elements

Let $R=(\mathbb{Z}/2\mathbb{Z})[t]$, $f=t^2+t+1$. Show that $R/(f)$ has four elements. So I know that $R/(f)$ has the form $\lbrace a+(f) \mid a \in R \rbrace$, and $(f)$ is the ideal generated ...
0
votes
2answers
17 views

Find out how many ring homomorphism exist

Consider $R=\mathbb{Z}_{11}\ \ $ $S=\mathbb{Z[\sqrt5]}=\{a+b\sqrt5: a,b \in \mathbb{Z} \}$ find out how many there is homomorpshism from $f:R\to S$ and $g:S \to R$ I think first ring isomorphsms ...
0
votes
1answer
33 views

Proving that a module homomorphism is an isomorphism if and only if the induced map between the localizations is an isomorphism

Let $P$ be a prime ideal of a ring $R$. Let $M$ and $N$ be $R$-modules, and let $f:M \rightarrow N$ be a module homomorphism. Let $f_P: M_P \rightarrow N_P$ be defined by $f_P \left(\frac{m}{s} ...
4
votes
1answer
67 views

If $e$ is idempotent and both $eRe$ and $(1-e)R(1-e)$ are orthogonally finite, is $R$ orthogonally finite?

Let $R$ be a ring with $1$ possessing a non-zero idempotent $e$. It usually happens that if $eRe$ and $(1-e)R(1-e)$ have a property $P$ then so does $R$. A ring is said to be orthogonally finite if ...
2
votes
1answer
87 views

Is $\mathbb Z$ first-order definable in (the ring) $\mathbb{Z\times Z}$?

Is $\mathbb Z$ first-order definable in $\mathbb{Z\times Z}$ (using sum and product but obviously not the concept of "component")? I believe no but how may I prove it? Is this standard?
0
votes
1answer
26 views

Representation of invertible elements in the Total ring of fractions [on hold]

Let $R$ be a ring and $S$ be the set of non-zero-divisor. And let $R_S$. If $\frac{x}{s}\in R_S$ is invertible then $\frac{x}{r}=\frac{y}{s}$ with $y,s\in S$? I thought that $\frac{x}{r}$ ...
4
votes
4answers
142 views

What is an example of a non-zero “ring pseudo-homomorphism”? [duplicate]

By "pseudo ring homomorphism", I mean a map $f: R \to S$ satisfying all ring homomorphism axioms except for $f(1_R)=f(1_S)$. Even if we let this last condition drop, there are only two ring ...
1
vote
1answer
38 views

By taking answers modulo $m$, show that $Z_m$ has no divisors of zero iff $m$ is prime

For a positive integer $m$, let $Z_m$ = {$0,1,...,m-1$} $+: Z_m * Z_m \rightarrow Z_m$ Mult: $Z_m * Z_m \rightarrow Z_m$ by taking answers modulo $m$ (e.g. $Z_6$ = {$0,1,2,3,4,5$} and $(3)(5)$ = ...
0
votes
1answer
30 views

There exists an injective ring homomorphism $\bar{\phi} : Q \to F $ such that $ j \circ \phi = \bar{\phi} \circ j$.

Let $A$ and $B$ two integral domains and $Q$ and $F$ their fields of fractions, respectively. Consider the ring homomorphisms $i : A \to Q$ and $j:B \to F$, defined as $i(a)=\frac{a}{1_A}$ and ...
1
vote
0answers
36 views

Monomorphism from a sheaf to a flasque sheaf: determining the stalk.

Let $M$ be a topological Hausdorff space. We use the following definitions (as they may vary): A presheaf $\mathcal{F}$ is a collection of vector spaces $\mathcal{F}(U)$ for each open subset $U$ of ...
0
votes
0answers
9 views

Posner, 1957 Derivations in prime rings

Let R be a prime ring, and d a derivation of R such that ad(a)-d(a)a=0. Then R is commutattive or d is zero. E.C.POSNER proved this lemma in his famous paper in 1957.but I can't understand how the ...
1
vote
2answers
50 views

Show that $x^5-x^2+1$ is irreducible in $\mathbb{Q}[x]$.

Show that $x^5-x^2+1$ is irreducible in $\mathbb{Q}[x]$. I tried use the Eisenstein Criterion (with a change variable) but I have not succeeded. Thanks for your help.