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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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.
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49 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 ...
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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 ...
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1answer
31 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 $k \in \mathbb{N}$, ...
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2answers
31 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 = ...
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1answer
65 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 ...
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1answer
36 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 ...
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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 $\mathcal{O}$ ...
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0answers
50 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 ...
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1answer
33 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 ...
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0answers
38 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 ...
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3answers
43 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 ...
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2answers
85 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 ...
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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 ...
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1answer
30 views

Integral ring extensions and finitely generated as a module

Let $A \subset B \subset C $ be rings. Suppose that $A$ is Noetherian and $C$ is finitely generated as an $A$-algebra. I want to show that $C$ is finitely generated as a $B$-module $ \iff $ $C$ is ...
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0answers
36 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) = ...
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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 ...
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3answers
99 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 ...
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3answers
58 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.
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1answer
54 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 ...
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2answers
72 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 ...
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2answers
49 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 ...
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2answers
142 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 ...
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3answers
200 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 ...
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1answer
36 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 $\mathbb{C}$. ...
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1answer
79 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 ...
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1answer
56 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 ...
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1answer
48 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.
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2answers
147 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: ...
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1answer
40 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 ...
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2answers
188 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 ...
2
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1answer
33 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 ...
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1answer
48 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 $$\frac{k[[T_1,\ldots,T_n]]}{(f)}$$ ...
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2answers
119 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} ...
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1answer
74 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 ...
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1answer
35 views

Let $K$ be a field extension of $F$ and let $a \in K$. Show that $[F(a):F(a^3)] \leq 3$

Let $K$ be a field extension of $F$ and let $a \in K$. Show that $[F(a):F(a^3)] \leq 3$. Find examples to illustrate that $[F(a):F(a^3)]$ can be $1,2$ or $3$. Attempt: $F \subset F(a^3) \subseteq ...
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0answers
45 views

A non-unital commutative ring with infinite elements such that each element $a$ satisfies $ab =0$ for infinitely many $b$'s

Beside the usual rules for non-unital commutative ring (that is, a ring without multiplicative identity) $R$, I want $R$ to satisfy the following: $ab = 0$ for each element $a$ and there are ...
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0answers
126 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) ...
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1answer
23 views

A field extension of prime degree

Suppose that $E$ is an extension of $F$ of prime degree. Show that $~~\forall~ a \in E : ~ F(a)=F$ or $F(a)=E$ Attempt: Suppose that $E$ is an extension of a field $F$ of prime degree, $p$. ...
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1answer
30 views

Suppose that $f(x)$ and $g(x)$ are irreducible over $F$ and that $\deg f(x)$ and $\deg g(x)$ are relatively prime.

Suppose that $f(x)$ and $g(x)$ are irreducible over $F$ and that $\gcd(~\deg g(x),\deg f(x)~)=1$. If $a$ is a zero of $f(x)$ in some extension of $F$, show that $g(x)$ is irreducible over $F(a)$ ...
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2answers
48 views

Space of matrices that commute with a given matrix

Let $A$ be an $n\times n$ complex matrix, and $C(A)$ be the vector space of all matrices that commute with $A$. I have to determinate if the dimension of $C(A)$ is greater or equal than $n$, or not. ...
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1answer
37 views

Krull's theorem for a ring that does not have unit (multiplicative identitiy)

Is there some sorts of Krull's theorem (that every ring has maximal ideal) for rings that do not have multiplicative identity (unit)? So I know that non-unital rings do not satisfy Krull's theorem, ...
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2answers
29 views

Nil radical of an ideal on a commutative ring

This is a problem of an exercise list: Let $J$ be an ideal of a commutative ring A. Show that $N(N(J))=N(J)$, where $N(J)=\{a \in A; a^n \in J$ for some $n \in \mathbb{N}\}$. What I did: ...
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2answers
62 views

Cohen-Macaulay and regularity

I know this is a simple question but to make sure....: $A$ is a commutative ring which is Cohen-Macaulay and for every maximal ideal $\mathfrak{m}$ in $A$ if $\dim A_{\mathfrak{m}}=\dim A$ then ...
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1answer
74 views

Counting diagonalizable matrices in $\mathcal{M}_{n}(\mathbb{Z}/p\mathbb{Z})$

How many diagonalizable matrices are there in $\mathcal{M}_{n}(\mathbb{Z}/p\mathbb{Z})$ ? Where $p$ is a prime number. Attempt : By definition a matrix is called diagonalizable if there exists an ...
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1answer
58 views

Showing the Weyl algebra is simple.

Let $R$ be a ring with $1$, which contains $\mathbb{Q}$, and generated over $\mathbb{Q}$ by two elements $x,y$ such that $yx-xy=1$. Show that $R$ is simple. What i did? Certainly $x, y \in R$ as ...
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1answer
35 views

Showing that $R/[R,R]$ is commutative

Let $R$ be a commutative ring with $1$, and let $[R,R]$ be the ideal generated by the set $$ \{ \ xy-yx \ : \ x,y \in R \ \} $$ Show that $R/[R,R]$ is commutative. Before I show what I did, I ...
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2answers
76 views

Ring of rational-coefficient power series defining entire functions

I'm wondering if anyone has come across the following ring before. Let $R$ be the ring of complex power series $f=\sum_{n \ge 0} a_n t^n$ such that $a_n \in \mathbb{Q} \: \: \forall \: n$ The ...
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2answers
67 views

Can $R \times R$ be isomorphic to $R$ as rings?

I know from this question that a ring $R \times R$ can be isomorphic to $R$, as $R$-modules. But can they ever be ismorphic as rings?
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2answers
39 views

Kahler differentials and quotient rings.

I am dealing with some nice rings that are always isomorphic to some fairly nice quotient ring of a polynomial ring. A typical example is: $$ \mathbb{C}[X,XY,XY^2] \cong ...