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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-3
votes
2answers
66 views

Idempotent elements of a ring.

I need the idempotent elements of $Z_{900}$ $2^2\cdot 3^2\cdot 5^2=900$ Of course there's $$0 \pmod 4 \\ 0 \pmod 9 \\ 0 \pmod {25} \\ $$ and $$ 1 \pmod 4 \\ 1 \pmod 9 \\ 1 \pmod {25} \\ $$ I found ...
0
votes
0answers
20 views

A non-UFD where there exist infinitely many elements such that $a^2 \mid b^2$ does not lead to $a\mid b$ [duplicate]

Is there a commutative non-$\text{UFD}$ ring such that there exists a set $X$ of infinite cardinality of elements that for $\forall x \in X$, $x^2$ is a multiple of $a^2$ for some particular $a$, but ...
0
votes
2answers
40 views

What would be a prime element in the field of rational numbers?

It is clear to understand what prime elements will be in case of ring of integers and many other rings. However, I find it confusing in case of fields, more specifically field of rational numbers. So ...
1
vote
1answer
32 views

Example of a module such that every proper submodule is finitely generated but the module is not.

Let $R$ be a ring with 1 and $M$ an $R$-module. What is an example such that $M$ is infinitely generated but every proper submodule is finitely generated.
0
votes
0answers
11 views

Let $R$ be an indecomposable ring with $(R,+)=(Z_2\oplus Z_2\oplus Z_2,+)$. If $|J(R)|=2$ and $R/J(R)=GF(2)\oplus GF(2)$, then $R$ has a unity.

Prove the following assertion: Let $R$ be an indecomposable ring with $(R,+)\cong (\Bbb{Z}_2\oplus \Bbb{Z}_2\oplus \Bbb{Z}_2,+)$. If $|J(R)|=2$ and $R/J(R)\cong GF(2)\oplus GF(2)$, then $R$ has a ...
5
votes
1answer
143 views

Lang's “General Integrality Criterion”

Theorem 3.7 in the chapter on ring extension on page 352 of the latest edition of Lang's "Algebra" appears redundant in its phrasing to me. Specifically, if $g_s$ is a polynomial of total degree ...
6
votes
1answer
58 views

A central division algebra is not its commutator

In looking at old qualifying exam questions, I've come upon a question that has me stumped. Let $A$ be a central division algebra (of finite dimension) over a field $k$. Let $[A,A]$ be the ...
3
votes
2answers
139 views

Sum and Product of two transcendental numbers cannot be simultaneously algebraic

If $\alpha$ and $\beta$ are real number and $\alpha$ and $\beta$ are transcendental over $\mathbb Q$, show that $\alpha \beta$ or $\alpha +\beta$ is also transcendental over $\mathbb Q$ Attempt: ...
1
vote
3answers
52 views

A number system that is not unique factorization domain

Can anyone present a number system that is not unique factorization domain and is a commutative ring? So I want the case that does not involve polynomials/monomials or some trivial cases.
0
votes
0answers
22 views

Question about valuation rings of a rational function field

Let $k$ be a field and $f \in k[x,y] \setminus k$. Write $E=k(f)$ and $L=k(x,y)$. Assume that there exists a $g \in k(x,y)$ such that $L = E(g)$. Suppose there exists a valution ring $\mathcal{O}$ ...
0
votes
1answer
27 views

Example of commutative algebra over integers where there exists $x$ such that $x = y^2$ for several $y$'s

Is there a commutative algebra over integers such that there exists $x$ with $x = y^2$ for several $y$'s? Also, is there a commutative algebra over integers such that for every $k \in \mathbb{N}$, ...
-1
votes
2answers
29 views

every finite integral domain is a field

I am trying to understand a proof that every finite integral domain is a field, and in part is states: "Consider $a, a^2, a^3,\dots$. Since there are only finitely many elements we must have $a^m = ...
7
votes
3answers
124 views

In $\mathbb{Z}/(n)$, does $(a) = (b)$ imply that $a$ and $b$ are associates?

[Update: Based on the hints provided by @zcn and @whacka, I believe I have found a solution. See my answer below.] Below, $R$ is a commutative ring with $1$. In John J. Watkins' Topics in ...
3
votes
0answers
32 views

Non-UFD that there exists set $X$ of any cardinality and $a$ that $xy$ is not divisible by $a^2$ for any $x,y \in X$ and $x^2$ is divisible by $a^2$

Let's say we want to construct a non-UFD that is a commutative ring that satisfies: There exists $a$ such that there exists a set $X$ of elements of finite cardinality $k$ such that for any $x,y \in ...
2
votes
1answer
104 views

Why is $\mathbb{F}_p[X]/(X^2+1)$ not a field if $p \equiv 1 \bmod4$

here I am again with another question. Assume that p is prime, and $ \ p \equiv 1\bmod4$. Prove that $\mathbb{F}_p[X]/(X^2+1)$ is not a field. I don't know how to tackle this problem. First I'll ...
1
vote
1answer
29 views

Ideal quotient and extension

Let $R$ be a commutative ring and $S$ a subring of $R$. If $I$ is an ideal of $S$ define $I^e$ as the ideal in $R$ generated by $I$, i.e. the extension of $I$ in $R$. If $I,J$ are ideals in $S$, we ...
0
votes
1answer
34 views

Relating the characteristic of the ring R to the characteristic of R[x]

Suppose $R$ is a ring and $R[x]$ is the ring of polynomials in the indeterminate $x$ with coefficients from $R$. The characteristic of a ring is the smallest positive integer $n$ such that $n \cdot r ...
0
votes
3answers
40 views

Converting a polynomial ring to a numerical ring

One may be curious why one wishes to convert a polynomial ring to a numerical ring. But as one of the most natural number system is integers, and many properties of rings can be easily understood in ...
1
vote
0answers
48 views

Rapid and easy question on ideals and ring

Let $R$ be the number ring related to a field $K$ of finite degree over $\mathbb Q$, i.e. $\mathbb Q\le K\le\mathbb C$ and $[K:\mathbb Q]=n$. Hence $R=\mathbb A\cap K$, where $\mathbb A$ is the ring ...
4
votes
2answers
76 views

A non-UFD such that $a^2 \mid b^2$ does not lead to $a\mid b$

Is there any non-UFD that is a commutative ring such that $a^2 \mid b^2$ does not always lead to $a\mid b$? It would be preferable if examples are something that does not involve ...
3
votes
2answers
94 views

Prove that $\mathbb{F}_5[X]/(X^2+3)$ is a field with 25 elements

Prove that the ring shown above is a field containing 25 elements. Research effort: $$\mathbb{F}_5[X]/(X^2+3)\cong (\mathbb{Z}[X]/5\mathbb{Z}[X])/ \overline{(X^2+3)} \cong \mathbb{Z}[X]/(5,X^2+3)$$ ...
3
votes
2answers
76 views

How to show this ring is not a UFD

I am trying to show that $R=\Bbb Z[x,y,z,w]/(xw-zy)$ is not a UFD. Let $I=(xw-zy)$. Let $X=x+I$, $Y=y+I$, $Z=z+I$, and $W=w+I$. My guess is that $X$ is irreducible and therefore $(X)$ is a ...
0
votes
3answers
67 views

Irreducible and not prime [duplicate]

You know a counterexample for: Let $R$ be a commutative ring ($1\neq{0}$). If $a\in{R}$ is irreducible then $(a)$ is prime. Thanks.
0
votes
0answers
28 views

Isomorphic matrix groups over rings

I've thinking about this problem for the last couple days and I can't get anywhere. I would really appreciate some help. Is it true that, a) $\operatorname{SL}_n(\mathbb{Z}/2013\mathbb{Z})\cong ...
0
votes
1answer
23 views

Noetherian Ring Question

Let $A \subset B \subset C $ be rings. Suppose that A is Noetherian and that C is finitely generated as an A-Algebra. I want to show that C is finitely generated as a B-Module $ \iff $ C is integral ...
1
vote
3answers
100 views
+50

Some practical questions on cohomology and the ring $\mathbf{Z}[x]/(x^2)$

So I know that the cohomology ring of $S^n$ is $\mathbf{Z}[x]/(x^2)$ with "$x$ in degree $n$"; if we ignore the grading then this ring fails to distinguish the spheres. What is actually meant by "in ...
1
vote
0answers
32 views

Showing the existence of sub fields of a finite field

For each divisor $m$ of $n$, $GF(p^n)$ has a unique sub field of order $p^m$ . The proof of this theorem in Gallian goes like this : Suppose that $m$ divides $n$. Then, since : $(p^n-1) = ...
3
votes
3answers
92 views

Is division allowed in rings and fields?

Is division allowed in ring and field? The definition of ring I am using here does not require the presence of multiplicative inverse. I think in general, division is not a well-defined ...
1
vote
1answer
22 views

Every prime ideal is max. if anysequence $I_1 \supseteq I_2 \supseteq I_3 \supseteq \cdots $ stabilises? [duplicate]

Let $R$ be a comutative ring with $1 \neq 0 $. Assume that for every sequence of ideals $I_1 \supseteq I_2 \supseteq I_3 \supseteq \cdots $ there is a $N$ such that $$ n,m \geq N \quad ...
5
votes
1answer
328 views

Krull dimension of Noetherian local rings is finite

Does anyone know an "elementary" proof of the fact that a Noetherian local ring has finite Krull dimension? The one I know is from Atiyah&Macdonald's book Introduction to Commutative Algebra, ...
7
votes
2answers
178 views

Showing that $x^3+y^3+z^3=0$ is not rational

Is there a short proof that $F:x^3+y^3+z^3=0$ in $\mathbf{P}^2$ is not rational, apart from using the genus? Perhaps this is an elliptic curve, so every morphism $\mathbf{P}^n\rightarrow F$ is ...
0
votes
1answer
52 views

Proof that presheaf is a sheaf for Spec

Atiyah Macdonald define presheaf (chapter 3, exercise 23) on the base of $Spec(A)$, where $A$ is commutative ring with $1$, as follows $$ \mathfrak{F}(X_f) = A_f, $$ where $X_f$ is a basic open set ...
0
votes
2answers
44 views

Let $R=\mathbb{Z}/2[x]$ and $I=R(x^{17}-1)$. Is there a nonzero element $y$ of $R/I$ with $y^2=0$?

Let $R=\mathbb{Z}/2[x]$ and the ideal $I=R(x^{17}-1)$. Is there a non zero element $y$ of $R/I$ with $y^2=0$? My approach: Suppose there is $y=\bar{f}(x)$ with $f(x)\in R$, then ...
1
vote
2answers
137 views

Every element in a ring with finitely many ideals is either a unit or a zero divisor.

I came across the above proposition on mathstackexchange If every nonzero element of $R$ is either a unit or a zero divisor then $R$ contains only finitely many ideals. the link asks a different ...
0
votes
1answer
35 views

A matrix lies in a subring isomorphic to $\mathbb{C}$

Problem: Consider the matrix $$A = \begin{pmatrix} 0 & 3\\ -4 & 1 \end{pmatrix}.$$ Show that $A$ lies in a subring of Mat$_{2\times 2}(\mathbb{R})$ that is isomorphic to $\mathbb{C}$. ...
8
votes
2answers
339 views

Does every ring with unity arise as an endomorphism ring?

I don't believe that every ring with a $1$ is the endomorphism ring of an abelian group but I currently don't see how to produce a counterexample.
1
vote
1answer
78 views

The spectrum of a commutative ring with unity and its “topology”

Let $\operatorname{Spec}(R)$ be the set of prime ideals in the commutative ring with unity $R$, and let $\mathfrak a$ be some ideal. Show that we get a topological space if we define the closed sets ...
2
votes
1answer
168 views

Does codimension equal height in complete local domains?

For an ideal $I$ in a commutative ring $R$, define $\operatorname{codim}I=\dim R-\dim R/I$. Does codimension equals height for all ideals in the formal power series ring? Does this hold for complete ...
1
vote
1answer
51 views

Question concerning a list sorting problem

I have the following question: Let $a,b,c,d$ be four natural numbers with $a \leq b$ and $c\leq d$. I have written a program that produces a list, which has as entries all 2-tuples $(x,y)$ with ...
2
votes
1answer
44 views

Von Neumann regular but not self-injective ring

I want an example of a von Neumann regular ring which is not self-injective. My thanks go to anybody answering.
1
vote
1answer
257 views

When is the localization of a module trivial?

Let $R$ be a commutative ring and $M$ a finitely generated $R$-module. Let $S^{-1}M$ be localization of $M$, where $S$ is a multiplicatively closed subset of $R$. How to show that $S^{−1}M =0$ if ...
5
votes
3answers
119 views

Can Boolean ring without unit be embedded into a boolean ring?

While going through a book (Lectures on Boolean algebra, Halmos) I got struck at the following question : Prove that every Boolean ring without a unit can be embedded in a Boolean ring with a unit. ...
1
vote
1answer
54 views

Characterizing maximal ideals in $\mathbb{Z}[x]$

I need to prove this: Let $I\subset\mathbb{Z}$ be the ideal generated by $\{p,f(x)\}$, with $p$ prime in $\mathbb{Z}$. Then $I$ is maximal iff $f(x)$ is irreducible modulo $p$. So I was trying to ...
5
votes
2answers
556 views

Interview preparation for Ph.D admission

I have recently gave a written exam for admission to Ph.D program to an institute in India. I have done that exam well and hoping for an interview call. I would like to know what could be type of ...
0
votes
0answers
101 views

Multiplicity of a module in a particular case

We define multiplicity of a module M of dimension $d>0$ as $$mult(M) := lc (P_M) (d-1)!$$ where $P_M$ denotes the Hilbert polynomial of M. Equivalently, we have $mult(M) = Q_M(1)$, where $HP_M (z) ...
1
vote
1answer
36 views

Characterizing Prime and Maximal Ideals in a nice Ring

Consider the "nice" ring $(\mathbb{Z}/20\mathbb{Z})[x]$ and I am trying to list all the prime and maximal ideals of this. The reason I call this a nice (or manageable) ring is because we ...
0
votes
1answer
46 views

A nonregular local ring [duplicate]

Consider the ring of the formal power series $k[[T_1,\ldots,T_n]]$ ($k$ algebraically closed) where $\mathfrak m$ is the maximal ideal. If $f\in\mathfrak m^2$, why $$\frac{k[[T_1,\ldots,T_n]]}{(f)}$$ ...
0
votes
1answer
71 views

Some residue field

Consider a prime ideal $\mathfrak{p}\in\mathrm{Spec} \ \mathbf{Z}[x]$; the residue field at $\mathfrak{p}$ is the fraction field of $\mathbf{Z}[x]/\mathfrak{p}$. Can we classify the residue fields? I ...
2
votes
1answer
32 views

Example of a ring which is not CM at all its prime ideals

A commutative ring $A$ is said to be CM at a maximal ideal $\mathfrak{m}$ if and only if $Depth(A_{\mathfrak{m}})=Krull(A_{\mathfrak{m}})$. What is an example of a connected commutative ring $A$ which ...
1
vote
2answers
117 views

Is there a more direct way of proving that this ring is an integral domain?

In self studying abstract algebra and I've come upon the following problem which I could not solve directly. For any $d\in \mathbb{Z}$ we are asked to show that $\mathbb{Z}[\sqrt d]=\{a+b\sqrt{d} ...