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

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4
votes
5answers
63 views

Examples of properties not preserved under homomorphism

An isomorphism indicates that two structures are the same, using different names for the elements. Therefore it's obvious that every (algebraic) property of the first structure must be present in the ...
1
vote
1answer
22 views

transitivity of integral extensions

Let $T{\geq}S{\geq}R$ be commutative rings. I'm trying to prove that if $T$ is integral over $S$ and $S$ is integral over $R$ then $T$ is integral over $R$. Let $t$ be in $T$ so there exist ...
2
votes
2answers
119 views

Localisation and prime ideals

If $A$ is a ring and $S=\{1,f,f^2,f^3,...\}$ a multiplicative set of $A$. Prove that $Spec(A_f)=(\mathfrak{V}((f)))^c$. Notation: $A_f=S^{-1}A$ and $\mathfrak{V}((f))=\{P \in Spec(A): P \supset (f)\}$ ...
1
vote
0answers
12 views

Functional equation on a ring

Let $R$ be a ring with identity. Find all functions $f: R \to \Bbb R$ satisfying the functional equation \begin{equation} f(xp+yq, yp+xq)=f(x, y)f(p, q) \end{equation} for all $x, y, p, q\in R$, and ...
-1
votes
0answers
11 views

Which of the following Statements are true(CSIR)

Question : Let $R$ and $S$ be non zero commutative rings with unity. Then which of the following statements are true If $S$ is a quotient ring of $R$, then either Char(R) divides Char(S) or ...
0
votes
1answer
26 views

$(p)$ is a prime ideal in $\mathbb{Z}[\sqrt{n}]$

Let $n$ be a square-free integer such that $n\equiv 0,2,$ or $3\bmod 4$. If $p\in\mathbb{Z}$ be a prime such that $n$ is not a square modulo $p$, then $(p)$ is a prime ideal in ...
1
vote
2answers
21 views

How to show $\mathbb{Z}[x]/<2,x>$ is isomorphic to $Z_2$

I'm having quite a bit of trouble figuring out why $\mathbb{Z}[x]/<2,x>$ is isomorphic to $\mathbb{Z}_2$. So far I have figured out there is an onto map $\zeta: ...
0
votes
2answers
58 views

Showing quotient rings are isomorphic

Can anyone explain to me how to show two quotient rings are isomorphic? For my particular case. Both quotient rings are based off ideals in the ring $\mathbb Z_3[X]$: $$ \mathbb ...
0
votes
0answers
10 views

Why an element of an order of a number field K is always an algebraic integer of K?

Let $K=\mathbb Q(\sqrt{N})$ be a number field, $\mathcal O$ be an order of $K$ (i.e. $\mathcal O$ is a subring of $K$ and $\mathcal O$ is a free $\mathbb Z$-module of rank 2). In the begining of ...
0
votes
1answer
25 views

Unique prime ideal implies every element is nilpotent or a unit.

Let $R$ be a commutative ring with only one prime ideal. I want to show that every element of $R$ is either a unit or nilpotent, or equivalently, that the nilradical is the unique maximal/prime ideal. ...
3
votes
1answer
44 views

Kernel of a morphism of regular rings.

Let $k$ be a field and $f: A \rightarrow B$ be a surjective ring morphism between smooth Noetherian $k$-algebras. By smooth I mean that the module of Kahler Differentials $\Omega_{A|k}$ is a ...
1
vote
2answers
21 views

A homomorphism induces a continuous map from ${\rm Spec}(A') \to {\rm Spec}(A)$.

Let $A, A'$ be commutative rings with $1 \neq 0$. Let $h : A \to A'$ be such that $h(1) = 1$. Then $f: {\rm Spec}(A') \to {\rm Spec}(A)$ defined by $f(\mathfrak{p}') = h^{-1}(\mathfrak{p}')$ is ...
1
vote
0answers
19 views

Module is free of finite rank $\implies$ submodule is free of finite rank?

Let $M$ be $R$-module, where $R$ is commutative ring with $1,$ and $N$ be submodule of $M.$ If $M$ is free of finite rank, so is $N \ ?$ Answer: False. Let $R=M=\mathbb{Z}_{6}$ and ...
1
vote
1answer
16 views

Polynomial ring problem

May I verify if my proof to this problem is correct? Let $p \in \mathbb{P}.$ For $x \in \mathbb{Z},$ let $\overline{x}$ be remainder of $x$ when divided by $p.$ Let $f(X)= \sum^{n}_{i=0}a_iX^i ...
2
votes
2answers
50 views

Irreducible components of $Spec(A) $

A topological space $X$ is called irreducible if given $A_{1}, A_{2} $ open sets $ \neq \emptyset $ then $A_{1} \cap A_{2} \neq \emptyset$. The maximal irreducible topological subspaces of $X$ are ...
0
votes
0answers
45 views

exact sequence proof [on hold]

Let $R$ be a commutative ring and $0\to L\to M\to N\to 0$ be a sequence of $R$ modules. Let $A$ be a multiplicativity closed subset of $R$ so that we can consider the corresponding localisation ...
0
votes
1answer
22 views

Annihilaor of a prime is non-zero

If we have a commutative noetherian ring how do we know that the annihilator of a prime ideal is always non-zero?
1
vote
1answer
29 views

A problem on non-commutative ring

Let $R$ be a non-commutative ring with $1$ and $a,b\in R$ such that $ab=1 \neq ba.$ Could anyone advise me on how to show there exists $c\in R-\{b\}$ such that $ac=1 \ ?$ Hints will suffice. Thank ...
0
votes
1answer
24 views

Eisenstein's criterion pf

I know that 'Eisenstein's criterion'. I know that pf of state "(NOT $p$|$a_{n}$), [$P|a_i$ for ($0\le i\le n-1$)], (NOT $p^2$|$a_0$)". I know regular way. but I hope to Second pf way. $\;$ $\;$ ...
0
votes
1answer
36 views

Formal Power Series Ring, Maximal Ideal

Let K be a field. Show that K[[x]] (the formal power series ring with coefficients in K) has a unique maximal ideal. Attempt at a solution: Let I $\subset$ K[[x]] with I $\neq$ K[[x]] and I= ($x$). ...
4
votes
2answers
19 views

If $M$ is Noetherian, then $R/\text{Ann}(M)$ is Noetherian, where $M$ is $R$-module

Let $M$ be a $R$ -module and $\text{Ann}(M)=\{r \in R: rm =0 , \forall m \in M\} .$ Suppose $M$ is Noetherian, could anyone advise me on how to prove $R/\text{Ann}(M)$ is also Noetherian? Hints ...
0
votes
0answers
40 views

How can $\{ a+b \sqrt d : a,b \in \mathbb Z \}$ be a subset of $\mathbb C$?

I got this question on one of my homeworks. However as far as I am aware, $R \subseteq \mathbb C$ only when $ d= -1$. Am I overlooking something that makes it possible in the general case, or can ...
0
votes
1answer
28 views

Noncommutative finitely generated algebras need not be noetherian

I would like to understand an example (of the title) given in the book "An Introduction to Noncommutative Noetherian Rings" by K. R. Goodearl, R. B. Warfield... On page 8, Exercise 1E, an example of ...
0
votes
0answers
14 views

Multilinear Polynomial - Book “Polynomial Identities in Ring Theory” author Louis Halle Rowen

Rowen gives an example on page 7 of his book Polynomial Identities in Ring Theory says: Remark 1.1.23: Suppose $f$ has "constant term $0$" (i.e., each monomial of $f$ has degree $> 0$). If ...
0
votes
1answer
32 views

$f(x)$ is irreducible and primitive over $\Bbb Q$, then $f(x)$ is irreducible over $\Bbb Z$

show that $f(x) (\in \Bbb Z[x])$ is irreducible and primitive over $\Bbb Q$, then $f(x)$ is irreducible over $\Bbb Z$. How pf it? I tried it. MY pf) Suppose that $f(x)$ is reducible over $\Bbb Z$. ...
1
vote
1answer
31 views

An induction proof on a version of “prime avoidance” from Atiyah-McDonald.

The proposition is 1.11 from the commutative algebra book. Let $\mathfrak{p}_1, \dots, \mathfrak{p}_n$ be prime ideals and let $\mathfrak{a}$ be an ideal contained in the union of those prime ideals. ...
3
votes
1answer
46 views

Online Finite Field Calculator

I need to find an online Finite Field calculator with the following operations: Inverse SqrRoot Mult Div I have found one a couple of days ago but lost the url, and cannot find it now. Any ...
0
votes
1answer
32 views

$V(\mathfrak{a} \cap \mathfrak{b}) = V(\mathfrak{a}) \cup V(\mathfrak{b})$ (Spectrum of a commutative ring)

Let $V(\mathfrak{a})$ be all ideals in ${\rm Spec}(A)$ that contain ideal $\mathfrak{a}$. Then $V(\mathfrak{a} \cap \mathfrak{b}) = V(\mathfrak{a}) \cup V(\mathfrak{b})$. $\mathfrak{p} \in$ RHS ...
0
votes
1answer
40 views

How to show $c-b\lt b-a$

The question: Let $G$ be an Arf semigroup and $a\lt b\lt c$ be three consecutive elements in $G$. How to show that $c-b\lt b-a$ and how to show that this is not necessarily the case for every ...
0
votes
1answer
21 views

Sylvester domains

I'm an undergrad mathematics student and I'd like to request some books about Sylvester domains. Specifically I'd like to understand the fact that not all modules are Sylvester domains. I just proved ...
1
vote
0answers
55 views

Global dimension of $\mathbb Q [x]$

I'm trying to show that the global dimension of $\mathbb Q [x]$ is 1. I have shown that $D(\mathbb Q [x]) \leq 1$ as follows. One can reduce to the case of showing that $$sup_{\{B\}}\; \text{pd}\; ...
0
votes
1answer
41 views

Questions on a proof of “All prime ideals of a Dedekind domain are invertible”

I tried to prove this theorem : All prime ideals of a Dedekind domain is invertible. i.e, For every prime ideal $\mathfrak{p}$ of Dedekind domain $R$, there exists $\mathfrak{p}^{-1} \subseteq ...
0
votes
1answer
36 views

$\Bbb Z[i\alpha]$ UFD's

I know that $\Bbb Z[i]$ and $\Bbb Z[\sqrt{-2}]$ are Unique Factorization Domains, and that $\Bbb Z[\sqrt{-6}]$ is not. I have two questions. I know that they may be difficult questions, so I only ask ...
0
votes
0answers
24 views

Proving that $V(R^*)=V(R)-1$

Let $R$ be a noetherian local ring with Jacobson radical $J$. Define $V(R)=dim J/J^2$ where $J/J^2$ is considered as a vector space over $R/J$. Now fix $x\in J/J^2$. If we then let $R^*=R/xR$ then I ...
-1
votes
3answers
48 views

Let $R$ be a PID and $I$ is a non zero proper ideal of $R$. show that if $R/I$ has no nonzero zerodivisor, then it is a field. [closed]

Let $R$ be a PID and let $I$ be a non-zero proper ideal of $R$. Show that if $R/I$ has no non-zero zerodivisor, then it is a field.
2
votes
2answers
39 views

Sub-modules of free modules

I'm going back through basic module theory notes, and I've come across a paragraph explaining that a sub-module of a finitely generated free module may not itself be free. In my course a free module ...
3
votes
1answer
37 views

Show: $\varphi\colon\mathbb{Z}_{mn}\to\mathbb{Z}_m\times\mathbb{Z}_n, k\mapsto (k\% m,k\% n)$ is a ring isomorphism for $m$ and $n$ relatively prim

Let $m\in\mathbb{Z}, n\in\mathbb{N}$. Then there exist unique elements $q\in\mathbb{Z}, r\in\mathbb{N}$ with $0\leq r<n$ and $m=qn+r$. We write $r:=m\% n$. Let $m,n\in\mathbb{N}$ be relatively ...
0
votes
1answer
60 views

Ring of fractions $S^{-1}A$ and localisation

I'd really appreciate if somebody could help me with the problem 6.4 Reid (Undergraduate commutative algebra), because I've been trying to get the solutions for days and I don't see it. (a) Give an ...
0
votes
1answer
24 views

Notation Question about Rings

If $S = \langle2\rangle$ is the ideal generated by $2$ in $\mathbb{Z}$, what does $S[x]$ represent?
3
votes
2answers
48 views

What's the point of defining left ideals?

I admit, I haven't gotten really far in studying abstract algebra, but I was always curious (ever since I saw a definition of an ideal) why is the notion of left-sided ideal introduced when we ...
0
votes
0answers
32 views

Show by explicit calculation that $\varphi\colon\mathbb{Z}\to\mathbb{Z}_n, m\mapsto m\% n$ is a surjective ringhomomophism

Consider $m\in\mathbb{Z}, n\in\mathbb{N}$. Then there exist unique elements $q\in\mathbb{Z}, r\in\mathbb{N}$ with $0\leq r<n$ and $m=qn+r$. We write $r:=m\% n$. Show by explicit calculation ...
3
votes
1answer
75 views

Are $R=K[a,b,c,d]/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$ and $K[x^4,x^3y,xy^3,y^4]$ isomorphic?

I saw somewhere that $R=K[a,b,c,d]/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$ and $K[x^4,x^3y,xy^3,y^4]$ considered the same. Is it true? Why? I'm a beginner so please answer in details
1
vote
0answers
18 views

Show that the solutions lifted by Hensel's lemma, are the only solutions in $\mathbb{Z} / p^k \mathbb{Z}$

Suppose that $f(x) \in \mathbb{Z}[X]$ is a polynomial to which we can apply Hensel's Lemma. Suppose the set $A$ is the set of solutions of $f(x)$ in $\mathbb{Z} / p \mathbb{Z}$. I'd like to show that ...
0
votes
4answers
34 views

Subrings Between Integers and Rationals

I'm trying to come up with an example of a ring that is bound strictly between the integers and the rational numbers, but I'm finding this construction very difficult. If anyone has any advice on how ...
1
vote
1answer
18 views

If $l(a, b, c) = l(a', b', c')$, then $(a, b, c) = (a ', b', c')k$ for some $k \in F$?

Let $F$ be a division ring. Define $l(a, b, c) = \{(x, y, z) \in F^3 : xa + yb + cz = 0\}$. Question: If $l(a, b, c) = l(a', b', c')$ is it true that $(a, b, c) = (a', b', c')k$ for some $ k \in F$? ...
0
votes
0answers
68 views

A book suggestion - Algebraic geometry. (Arf rings and Hilbert functions)

I am studying algebraic geometry and I need to learn Arf rings and Hilbert functions. Please suggest me books / lecture notes... etc that explains this topic in detail. Thank you.
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votes
1answer
52 views

ring and module problem

Let $$F=\mathbb{R}$$ $$V=\mathbb{R}^{4}$$ consider two matrices $$S1=\begin{vmatrix} 0&-1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& -1\\ 0&0 & 1 & 0 ...
1
vote
1answer
35 views

Isomorphism between rings

Let $R$ be the ring of real valued continuous functions defined on the interval $[0, 1]$. Let $I = \left\lbrace f \in \mathbb{R} : f^2(0) + f^2(1) = 0 \right\rbrace$. 1) Prove that $I$ is an ideal. ...
1
vote
3answers
28 views

Show that if $m\in M_n$ and $k \in \Bbb Z_n$ then $mk\in M_n$.

Let $M_n = \{a\in \Bbb Z_n\mid \text{ there exists a non-zero integer $k$ with the property that }a^k\equiv 0 \pmod n \}.$ Show that if $m \in M_n$ and $k \in \Bbb Z_n$, then $mk\in M_n$. For which ...
3
votes
0answers
52 views

$p(x)=x^4-2x^2-4$ is irreductible over $\mathbb Q.$

I need to show that $p(x)=x^4-2x^2-4$ is irreductible over $\mathbb Q.$ Here's what I've done: Please tell me if it's correct Over $\mathbb C,$ $x^4-2x^2-4\\=(x^2-1)^2-5\\=(x^2-1+\sqrt ...