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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9
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4answers
583 views

Spectrum of $\mathbb{Z}[x]$

Can someone point me towards a resource that proves that the spectrum of $\mathbb{Z}[x]$ consists of ideals $(p,f)$ where $p$ prime or zero and $f$ irred mod $p$? In particular I remember this can be ...
1
vote
1answer
423 views

Can someone explain this non-noetherian subring example?

I'm trying to find an example showing that subrings of noetherian rings are not necessarily noetherian. So I just searched the net and this site, and came upon this: "A common example showing that a ...
1
vote
0answers
48 views

Having difficulty understanding cosets. [duplicate]

Possible Duplicate: What is a quotient ring and cosets? I am taking an algebra class and having trouble understanding the elements (cosets) and ideals of quotient rings. Could someone ...
1
vote
1answer
110 views

Parity of idempotent elements of a ring

Given that the ring R with unity has finite number of idempotents. How do we show that the number of idempotents is even?
3
votes
5answers
656 views

Number of ring homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{28}$.

Question: Find the number of non trivial ring homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{28}$. ($f$ is not necessarily unitary, i.e., $f(1)$ need not be $1$.) Suppose $f$ is a ring ...
1
vote
1answer
162 views

Let $L/K$ be a field extension. Must a $K$-homomorphism $\theta: L\rightarrow L$ be an isomorphism?

I am thinking about $\mathbb{C}/\mathbb{Q}$. But other than the identity map and taking complex conjugates, I cannot think of anything. Any ideas? Thanks. Edit I just managed to show that if $L/K$ is ...
1
vote
1answer
50 views

Let $a, b\in L\supset K$ be transcendental over $K$.

Show that $b$ is algebraic over $K(a)$, the subfield of $L$ generated by $K$ and $a$, if and only if $a$ is algebraic over $K(b)$.
5
votes
1answer
100 views

Global dimension of quasi Frobenius ring

Let $R$ be a quasi-Frobenius ring (so $R$ is self-injective and left and right noetherian). I want to prove that $lD(R)=0$ or $\infty$, where $lD(R)$ denotes the left global dimension. I'm ...
1
vote
1answer
284 views

Properties of commutative ring and ideal related problem

For a commutative ring $\mathbb{M}$ and ideal $\mathbb{A}$, let $N(A)$={x in M|there exists a non-negative integer $ n $ such that $x^{n}$ in $\mathbb{A}$}. Which of following is true for $N(A)=A$? ...
4
votes
2answers
190 views

sufficient conditions for being a PID

Let R be a commutative ring with identity. If every ideal generated by two elements of R is principal, then can we conclude that R is a PID? Also, if every finitely generated ideal of R is principal, ...
1
vote
1answer
480 views

Subrings generated by a set

I have a misunderstanding with subrings generated by a set. (The intersection of all subrings of R containing a set X and a subring R0 of R is the subring generated by X over R0 in the ring R). ...
5
votes
0answers
227 views

System of polynomial equations over rational field

Fix $n\geq 2$. Let $p:=x_1^2+\ldots+x_{n-1}^2+1\in\mathbb{Q}[x_1,\ldots,x_{n-1}]$. Suppose $u_1,\ldots,u_n,v_1,\ldots,v_n\in\mathbb{Q}[x_1,\ldots,x_{n-1}]$ satisfy the following equations: ...
1
vote
0answers
253 views

Question about the residue field of a localization

Let $k$ be an algebraically closed field and $A$ a finitely generated $k$-algebra which is an integral domain. Let $m$ be a maximal ideal of $A$. Does the proof that $A_m/mA_m$ (i.e. the residue field ...
1
vote
1answer
468 views

Prove that f(x) divides $x^{p^n} - x $ iff deg(f(x)) = $d$ divides $n$

I believe that I have the backward direction covered: Let $d|n$ say $n=dq$ for some $q$ in $\mathbb{F_p}[x]$. Consider the field $\displaystyle\mathbb{F_p[x]}/(f(x))$ which has $p^d$ elements. Take an ...
1
vote
1answer
120 views

What about the Cauchy-Frobenius-orbit-counting formula

I know the proposition that says: Let $\lambda$ be a homomorphism from a finite group $G$ into $\mathbb{C}^{\times}$. Suppose that $G$ acts on some finite set $\Omega$ and let $M$ be the number of ...
5
votes
1answer
214 views

Polynomials in matrices with integer entries

I'm looking for references, if there is any, for this problem: Characterize all elements $a \in M_n(\mathbb{Z})$ for which we have $\mathbb{Q}[a] \cap M_n(\mathbb{Z})=\mathbb{Z}[a].$ Here, by ...
0
votes
1answer
201 views

Projective module over a ring

If $R$ is domain, as a projective module always exist over R. But how to produce such a module over $R$.
2
votes
1answer
79 views

Confusion regarding what kind of isomorphism is intended.

For a class I'm taking this semester, I was given this question: (To guarantee clarity, I am quoting the full question even though my own question is only regarding the final sentence.) Let $G$ be ...
3
votes
3answers
347 views

Show $m^p+n^p\equiv 0 \mod p$ implies $m^p+n^p\equiv 0 \mod p^2$

Let $p$ an odd prime. Show that $m^p+n^p\equiv 0 \pmod p$ implies $m^p+n^p\equiv 0 \pmod{p^2}$.
0
votes
1answer
187 views

The associated primes of the dual

I was trying to prove the following formula: $\mathrm{Ass}\;M^*=\mathrm{Ass}\;R\cap\mathrm{Supp}\;M$ (we can suppose $R$ noetherian, $M$ finitely generated. If it is useful even $R$ local) I ...
0
votes
3answers
72 views

Conjugation of polynomial in $\mathbb{Z}[x]$.

Let $a,b \in \mathbb{Q}$ and $d \neq 0,1$ be a square free integer. Define $\overline{a + b\sqrt{d}} = a - b\sqrt{d}$ If $f \in \mathbb{Z}[x]$ show that: $f(\overline{\alpha}) = ...
2
votes
1answer
174 views

Number Ring, Dedekind = Intersection Ring of Integers by inverting primes

$R$ is a number ring, $K$ the field of fractions and $R$ is Dedekind. $\Rightarrow$ exists a set of primes $S$ of $\mathscr{O}_{K}$ and: $R=\bigcap\limits_{\mathbb{p} \notin ...
3
votes
2answers
219 views

Does every non-commutative ring have the same numbers of left ideals and right ideals?

We know that a non-commutative ring may have different numbers of left ideals and two-sided ideals. For example, a matrix ring over a field has only 2 two-sided ideals but it have some non-trivial ...
2
votes
1answer
101 views

Module over a ring with identity versus module over a ring without identity

I'm taking a course which introduces modules this semester. In the notes, it gives the definition of a module (or left module) over a ring (where we generally always assume rings have multiplicative ...
1
vote
1answer
328 views

Normalization of a Ring

What is the exact definition of a normalization of a Ring? I have to show this: normalization of multiplicative subset of domain And the answer already helped, but I don't know what $S^{-1}R'$ is ...
4
votes
3answers
168 views

Is there an alternate definition for $\{ z \in \mathbb{C} \colon \vert z \vert \leq 1 \} $.

Is there a method of constructing a subset of a reasonably arbitrary ring so that when the construction is applied the $\mathbb{C}$ the result is $B = \{ z \in \mathbb{C} \colon |z| \leq 1 \} $? My ...
3
votes
2answers
492 views

Show that any prime ideal from such a ring is maximal.

Let R be a commutative ring with an identity such that for all $r\in$ R, there exists some $n>1$ such that $r^n = r$. Show that any prime ideal is automatically maximal. Any hints?
0
votes
2answers
59 views

Is $t^{2}$ a prime element of $\mathbb{F}_{2}(t^{2},s^{2})$?

I wish to find out if $t^{2}$ is a prime element of $\mathbb{F}_{2}(t^{2},s^{2})$ so I can justify the use of Eisenstein on the polynomial $x^{2}-t^{2}\in\mathbb{F}_{2}(t^{2},s^{2})[x]$ I believe ...
0
votes
2answers
64 views

Irreducibility of $x^{3}-t\in\mathbb{C}(t)[x]$

Denote $F=\mathbb{C}(t)$ and consider $p(x)=x^{3}-t\in F[x]$ Is it true that $p$ is irreducible over $F$ ? My thoughts: I think that since it is not true that $t^{2}\mid t$ (I don't know how to ...
3
votes
2answers
124 views

jacobson radical of a module

Let $B:=k[r,s]/(r^2,s^2,rs)$ be a polynomial ring , $k$ a field and $m={\ }_BB$. What is the radical of $m$? Thanks for the help.
2
votes
3answers
364 views

Modules with projective dimension $n$ have not vanishing $\mathrm{Ext}^n$

Let $R$ be a noetherian ring and $M$ a finitely generated $R$-module with projective dimension $n$. Then for every finitely generated $R$-module $N$ we have $\mathrm{Ext}^n(M,N)\neq 0$. Why? By ...
0
votes
2answers
48 views

Expressing $I/J$ in terms of quotients of the larger ring

Let $R$ be a Noetherian ring. Let $I\subseteq J$ be nonzero left ideals of $R$. Can the factor ring $I/J$ be expressed in terms of sums, quotients or submodules of rings of the form $R/K$, where $K$ ...
1
vote
0answers
132 views

normalization of multiplicative subset of domain

I am stuck with this: Let $R$ be a domain with normalization $R' \subset K$. Show that for every multiplicative subset $S \subset R$, the normalization of $S^{-1}R$ equals $S^{-1}R'$. How do you ...
1
vote
1answer
66 views

Equivalent properties on the vanishing of Bass numbers

I'm studying on this notes. I'm finding some difficulties on proposition 12 on page 15. Let me recall what we are trying to prove: At first we are trying to prove that if inj ...
5
votes
3answers
475 views

fields are characterized by the property of having exactly 2 ideals [duplicate]

Possible Duplicate: A ring is a field iff the only ideals are $(0)$ and $(1)$ Michael Artin's Algebra in the introduction of maximal fields, there was a sentence stated that fields are ...
1
vote
1answer
494 views

Maximal ideals in matrix rings

Let $F$ be a field, $R$ the ring of matrices over $F$. I am running into an apparent contradiction with regard to the maximal ideals of $R$. On one hand, we know that $R$ is simple, so its Jacobson ...
2
votes
1answer
201 views

Why does this module have finite length?

Let $A$ be a noetherian local ring with maximal ideal $m$. Let $p$ be a prime ideal such that if $B=A/p$, then $\mathrm{dim}\;B=1$. Take $x\not\in p$, $x\in m$, and set $C=B/xB$. Then $C$ has finite ...
5
votes
1answer
491 views

Irreducible Components of the Prime Spectrum of a Quotient Ring and Primary Decomposition

Recently I encountered a problem (the first exercise from chapter four of Atiyah & McDonald's Introduction to Commutative Algebra) stating that if $\mathfrak{a}$ is a decomposable ideal of $A$ (a ...
2
votes
2answers
1k views

Coprime ideals in commutative unit ring [duplicate]

Possible Duplicate: Comaximal ideals in a commutative ring Let $I$ and $J$ are coprime ideals in commutative unit ring $A$. Is it true that $I^m$ and $J^n$ are also coprime for any ...
4
votes
1answer
67 views

Decomposition of the injective hulls

I think this question is easy but I just cannot see how to solve it. Let $R$ be a ring and $M$ an $R$-module. Suppose $0=\bigcap_{i=1}^n N_i$ is a decomposition of $0$ with irreducible submodules of ...
2
votes
1answer
49 views

Colimits of cosimplicial rings

The forgetful functor from the category of rings to the category of sets does not preserve arbitrary colimits. But it does preserve filtered colimits. For example, the colimit of a sequence of rings ...
0
votes
1answer
152 views

On the injective dimension of a module

Let $R$ be a ring and $M$ an $R$-module then inj dim $M\leq i\in\mathbb{N}$ if and only if $\mathrm{Ext}^{i+1}(N,M)=0$ for every cyclic module $N$. The implication from left to right is obvious, I'm ...
2
votes
1answer
81 views

When the injective hull is indecomposable

Let $R$ be a ring and $M$ an $R$-module. Denote by $E(M)$ the injective hull of $M$. I was trying to prove that the following conditions are equivalent: 1) $(0)$ is meet-irreducible in $M$; 2) ...
3
votes
2answers
270 views

Intersection of a subring and an ideal

Given a unital ring $R$ and its unital subring $P$ (with the same unit). Also, given a maximal left ideal $L$ of $R$. Must $L\cap P$ be a maximal left ideal of $P$? I don't think so, but are there any ...
6
votes
3answers
597 views

Does every prime ideal in a ring arise as kernel of a homomorphism into $\mathbb{Z}$?

Let $R$ be a commutative ring. Clearly the kernel of $h$ is a prime ideal whenever $h : R \rightarrow ...
4
votes
1answer
137 views

Why are projective modules contained in this class of modules?

Suppose $A$ noetherian and define $G(A):=\{M: M$ is an $A$-module reflexive and Ext$^i_A(M,A)=$Ext$^i_A(M^*,A)=0$ for $i\geq1\}$ Why are projective modules contained in this class? Of course if ...
1
vote
2answers
311 views

When are quotient maps induced by equivalence relations surjective and injective?

Let $\sim$ be an equivalence relation on a set $X$. Also, there is a natural function $p:X\to \tilde X$ where $\tilde X$ is a set of all equivalence classes. (Equivalence classes are defined as, ...
1
vote
2answers
1k views

proof of chinese remainder theorem for ring

Let $R$ be a ring(not necessary have "1") and let $I,J$ be ideals of $R$ such that $I+J=R$. I want to prove that there is a $x\in R$ such that $$x\equiv r ({\rm mod} I) \quad x \equiv s ({\rm mod} J) ...
3
votes
0answers
99 views

Applications of Govorov-Lazard Theorem?

The Govorov-Lazard Theorem states that a (right) module over an unital ring is flat iff it is a direct limit of finitely generated free (right) modules. I wonder if this theorem has interesting ...
13
votes
3answers
527 views

Pathologies in module theory

Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar ...