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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2
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1answer
68 views

$g(x) | f(x)$ show that $(f(x)) \subset (g(x))$

I have been given a problem recently that has been puzzling me for some time. The problem states If $g(x), f(x)$ are elements of a polynomial ring $F[x]$ and $g(x) | f(x)$ show that $(f(x)) \subset ...
3
votes
1answer
123 views

Cyclic group generator and multiplicative identity of correspondng ring

Can cyclic groups made into ring with unity such that multiplicative identity is not any generator? (Or does there exist example of one such cyclic group?) Can we make $(\mathbb{Z}, +)$ into ring ...
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2answers
43 views

What does an ideal generated by a subset look like?

I am acquainted with principal ideals relatively well but I am struggling to understand the more general definition of ideals generated by an arbitrary subset of a ring. I suppose my question comes ...
0
votes
1answer
25 views

Show that $V(\bigcup_{i \in I} E_{i})=\bigcap_{i \in I} V(E_{i})$

This is a part of a problem in Atiyah's Introduction to Commutative Algebra introducing the Zariski Topology. Here we are given that $(E_{i})_{i \in I}$ is a family of subsets of a unital commutative ...
2
votes
1answer
108 views

Maximal ideal in local ring

The maximal ideal in $\mathbb{Z}_{(2)}$ should be $(2)$, but I don't understand this well. Suppose I take $\frac35\in \mathbb{Z}_{(2)}$. It is not in $(2)$ but in $(3).$ But what is the ideal between $...
2
votes
2answers
129 views

Infinite rings with lots of zero divisors

Today I was trying to find an infinite ring $R$ whose all nonzero and nonidentity elements were zero divisors and actually found one: $\mathcal R =\text{Fun}(\mathbb N, \mathbb Z/2\mathbb Z)$. Given a ...
1
vote
1answer
20 views

Let $K$ be a field, $A \subset K$, and $p \subset A$. Then $\exists$ a valuation ring $R$ satistfying…

I was stuck when reading a proof of the following theorem (Matsumura p. 72-3, Theorem 10.2), Let $K$ be a field, $A \subset K$ a subring, and $p$ a prime ideal of $A$. Then there exists a ...
2
votes
1answer
442 views

Show the union of two subrings is generally not a subring

Show that the union of two subrings is a subring if and only if either of the subring is contained in the other. I have no trouble in going from right to left but cannot seem to be able to go from ...
1
vote
1answer
41 views

Simple and semi-simple over $\mathbb{Z}$

What is a necessary and sufficient condition on an integer $n$ for $\mathbb{Z}/n \mathbb{Z}$ to be simple as a module over $\mathbb{Z}$? Semisimple? In the case of simple I think that because it is a ...
0
votes
1answer
75 views

Positive Real Numbers forming a subring

I was wondering if the subset of positive real numbers forms a subring of the real numbers under the regular operations of addition and multiplication. My thought so far is that since 1 is clearly in ...
5
votes
2answers
236 views

A relation between the Jacobson radicals of a ring and those of a certain quotient ring

Let $R$ be a ring $J(R)$ the Jacobson radical of $R$ which we define for this problem to be all the maximal left ideals of $R.$ I'm trying to prove the following proposition with only the definition (...
3
votes
2answers
228 views

Ring with many one-sided zero-divisors

Does there exist a ring all of whose elements are left zero-divisors but only one element is a right zero-divisor? The motivation for asking this question is that if there exists atleast one left ...
4
votes
1answer
70 views

Let R be a ring with unity $1_R$ and let S (with unity $1_S$) be a subring of R. Prove that either $1_S=1_R$ or $1_S$ is a zero divisor of R.

Let R be a ring with unity $1_R$ and let S (with unity $1_S$) be a subring of R. Prove that either $1_S=1_R$ or $1_S$ is a zero divisor of R. My attempt: Let $a \in S$. Then $1_S*a = a$ and this a ...
0
votes
1answer
302 views

Vector spaces and multiplicative inverse?

Do vector spaces have multiplicative inverses? They seem to be monoids under $+,\times$, so monoids $(\Bbb F, +)$ and $(\Bbb F, \times)$ where $\Bbb F=\Bbb R \,or\, \Bbb C$ And it is even a group ...
5
votes
2answers
338 views

Example of a Non-Commutative Division Ring With Finite Characteristics

Wedderburn's Little Theorem says that every finite Division Ring is commutative. What is about an infinite Division Ring with prime characteristics? Is this also a Field?
1
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2answers
722 views

Ideals of formal power series ring

I need help understanding the following solution for the given problem. The problem is as follows: Given a field $F$, the set of all formal power series $p(t)=a_0+a_1 t+a_2 t^2 + \ldots$ with $a_i \...
1
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2answers
114 views

Why doesn't $xa = x$ for all $x \in R$ imply that $a$ is the unit of $R$?

We have a ring $R$ as follows: Why is it not enough to conclude that $a$ is a unity if $xa = x$ for all $x$ in $R$? Is it because it is by definition that the unity satisfies $ax = xa = x$ for all ...
1
vote
1answer
66 views

Does Euclidean division not work for general polynomials?

If $K$ is a field. Then in $K[X]$ there is an Euclidean algorithm and if $K$ is replaced by any arbitrary commutative ring $R$, then almost we have an Euclidean algorithm, by the following result: ...
1
vote
1answer
39 views

Commutative matrix question

I was doing my HW, and I am confused with one thing. To show that a matrix is commutative, do we need to show both $x+y = y+x$ and $xy=yx$? Or just by showing $xy=yx$ would suffice?
3
votes
1answer
65 views

Why we throw away the units in the definition of irreducible elements?

In the book "Abstract Algebra" by Dummit, the definition of irreducible element in an integral domain $R$ goes like this. Suppose $r\in R$ is nonzero and is not a unit. Then $r$ is called irreducible ...
3
votes
1answer
81 views

Can every group be extended to ring with idenity [duplicate]

Can every abelian group converted into ring(by defining multiplication operation) with identity with same order. We can convert every group G into ring by defining a.b = 0 for all a and b in G. But ...
1
vote
1answer
35 views

Two sets of polynomials with distinct roots build the ring of polynomials.

Definitions: $i \in K$ $U_{i}:=\{f\in K[X] |f(i)=0 \}$ $K[X]$ is the ring of polynomials HINTS: K[X] is a vector space Every $U_{i}$ is a vector subspace of $K[X]$ Question: (i) With $s \neq ...
0
votes
3answers
222 views

Prove that $a+ib$ is prime in $\Bbb Z[i]$, if $a^2+b^2$ is prime in $\Bbb Z$.

Prove that $a+ib$ is prime in $\Bbb Z[i]$, if $a^2+b^2$ is prime in $\Bbb Z$. My Try: We can easily show that $\Bbb Z[i]$ is a FD but how can we show that $\Bbb Z[i]$ is a UFD. Because if we can show ...
0
votes
2answers
105 views

Show that $2$ is not prime in $\mathbb Z[\sqrt{-d}]$ for odd prime $d$

I have to show that if $d > 2$ is a prime number, then $2$ is not prime in $\mathbb Z[\sqrt{-d}]$. The case when $d$ is of the form $4k+1$ for integer $k$ is quite easy: $2\mid (1+\sqrt{-d})(1-\...
1
vote
1answer
77 views

how does Macaulay2 computes analytic spread for non-local rings?

Macaulay2 computes analytic spread for R=QQ[a,b,c,d,e,f] which is not a local ring. In the books like ...
1
vote
2answers
78 views

Criteria for the existence of zero-divisors and idempotent elements in the integers modulo $n$

I need help in establishing or at least deciding the validity of the following two criteria: There are in the ring $Z_n$ non-trivial zero divisors if only if $n$ is divisible with some square. ...
1
vote
0answers
227 views

How to solve this algebra problem?

Let $e$ be the idempotent element of the ring R. If $\langle e\rangle$ is the principal ideal generated with $e$, show that $R\simeq\langle e\rangle\times A(\{e\})$. I think $A$ s ring which contains ...
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vote
3answers
41 views

I need help to solve this problem

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 $J \subseteq I$.
0
votes
1answer
57 views

Automorphism group of the ring $\mathbb{F}_3\left[t,\frac{1}{t}\right]$

Let $R=\mathbb{F}_3\left[t,\frac{1}{t}\right]$ be a ring. What is the simplest form of $\mathrm{Aut}(R)$ ? Here $t$ is a variable and $R$ is the smallest ring contained in field $\mathbb{F}_3(t)$...
1
vote
1answer
79 views

Properties of Jacobson radical

I'm looking for an example of ring epimorphism $\varphi:R\rightarrow S$ such that the natural homomorphism $\tilde\varphi:J(R)\rightarrow J(S)$ is not surjective, where $J(R)$ is a Jacobson radical.
0
votes
1answer
40 views

Generators over semiperfect rings

It is clear that if $R$ is a ring with identity and $e\in R$ is an idempotent then $Re$ is a direct summand of $R$ while $R$ is a generator in the category of left $R$-modules. I have my question when ...
4
votes
1answer
104 views

Prove that $(2)$ is a prime ideal in $\mathbb Z[w]$

Let $w\in\mathbb C$ be such that $w^3=1$ and $w\neq1$. Prove that $(2)$ is a prime ideal in $\mathbb Z[w]$, and describe $\mathbb Z[w]/(2)$. What I wanted to do is to show that $\mathbb Z[w]$ is a ...
2
votes
2answers
103 views

Verify that R is a ring

Let $\alpha = \frac{1}{2}(1+\sqrt{-19}) \in \mathbb{C}$ and $R = \{a+b\alpha\mid a,b \in \mathbb{Z}\} \subseteq \mathbb{C}$. Let, $x = (a+b\alpha), y = (c+d\alpha)$ (I am good with showing $x - y \in ...
3
votes
3answers
66 views

Simple form of a ring

What is a simple form of this ring: $$\mathbb{Z}[\sqrt{2}][x]/(5,x^2+1),$$ I know that $\mathbb{Z}[\sqrt{2}][x]=\mathbb{Z}[x,y]/(y^2-2)$. Probably, I should use second theorem of isomorphism, but I ...
3
votes
0answers
71 views

Rings with bounded index of nilpotency are Dedekind-Finite

Recently in an article by A. A. Klein I have seen this result: A ring $R$ with bounded index of nilpotence is Dedekind-Finite. Can anyone help me prove this result?
-1
votes
1answer
62 views

Is $\{3,5,6\}$ a $2\mathbb{Z}$-sequence?

I have just started reading about the concept of $M$-regular sequences on my own and to understand the definition I asked myself the following question: Is $\{3,5,6\}$ a $2\mathbb{Z}$-sequence? ...
3
votes
1answer
55 views

What happens if we change the definition of quotient ring to the one that does not have ideal restriction?

From Wikipedia: Given a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows: a ~ b if and only if a − b is in I. ...
1
vote
1answer
74 views

Example of $I$-adic topology of submodule not matching subspace topology?

I'm reading about the $I$-adic topology on $M$ for $R$ a commutative ring, $I$ an ideal of $R$ and $M$ an $R$-module. The references I'm reading don't provide examples, but they say that if $N$ is a ...
2
votes
3answers
147 views

Show that $R/I$ is a field, where $R$ is a PID , where $I$ is a nonzero prime ideal.

Let $I \neq \{0\} $ be a proper ideal of a $PID$ $R$ such that the quotient ring $R/I$ has no zero divisors. I have a problem in showing that $R/I$ is a field. Help Needed!!
2
votes
2answers
39 views

Show that $q\equiv_8 1$ when $q$ is an odd square number [duplicate]

Problem: Given: q is an odd squared number - show that: $q\equiv_8 1$ My assumption: $\forall q\in N:\exists a \in Z: a =1\pmod{2}$ and $a^2=q$. Then I tried to show that it's only true satisfyingly ...
4
votes
2answers
64 views

Why is $I$ often an ideal in quotient ring $A/I$?

When talking about quotient ring $A/I$, where $A$ is a ring, $I$ is often assumed to be an ideal. Why is this so? What makes ideals very important when discussing quotient ring?
3
votes
2answers
72 views

Suppose $M$ is a Noetherian $R$-module and $\phi: M \rightarrow M$ is an $R$-module endomorphism of $M$.

So I did an exercise in my algebra textbook which was to show that $\ker(\phi^n) \cap \operatorname{im}(\phi^n) = 0$ and show that if $\phi$ is surjective, then $\phi$ is an isomorphism. I thought to ...
1
vote
1answer
59 views

For a commutative ring $R$, why does $1-ab$ being a non-unit leads to $1-ab \in M$ for some maximal ideal $M$?

Suppose there is a commutative ring $R$, without any restriction. Now suppose $a,b \in R$. If $1-ab$ is a non-unit, why is there at least one maximal ideal $M$ that $1-ab \in M$?
2
votes
1answer
65 views

Is $(X^3 - 18X + 12, 5) \in \mathbb{Z}[X]$ a prime ideal?

I'm trying to determine wheter $A = (X^3 - 18X + 12, 5)$ and $B = (X^3 - 18X + 12, X-1)$ is a prime ideal in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. I know that $A = \mathbb{Q}[X]$ since I can make $\...
2
votes
1answer
37 views

Is it always possible to extend a ring to a unital ring?

Just started learning algebra. So it's defined that ring is the ring not requiring a multiple 1, while unital ring does. Given a ring, is it always possible to extend it to a unital ring?
1
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1answer
53 views

Question related to commutative ring being Noetherian

Let $A$ be a commutative ring with $1$, and $A = (f_1, \ldots, f_n)$. I want to prove the following: If $A$ is a Noetherian ring, then so is $A_{f_i}$ (which is the ring $A$ localized at $f_i \in ...
1
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1answer
69 views

Is ring R itself a finitely generated module over $R$?

It seems trivial that ring $R$ itself is a $R$-module. But then can we say R is finitely generated by multiplicative identity? That seems so trivial..
2
votes
1answer
135 views

the number of zero divisors in polynomial ring

I was looking for an answer on the question How much zero divisors are in the ring $\dfrac{\mathbb{Z}_3[x]}{(x^4 + 2)}$? when I came up with the brilliant/hack-isch idea that it might just be $...
1
vote
2answers
171 views

Total ring of fractions of a Noetherian reduced ring is artinian

I'm doing the preparation to an exam, and I'm stuck in the following: If $R$ is a Noetherian ring with zero nilradical ($N(R) = 0$), and $S$ is the set of regular elements of $R$ ($r \in S$ if $rs ...
3
votes
1answer
40 views

Integral closed domain and localization of $\mathbb{Z}$ respect to prime ideal

We know that $\mathbb{Z}$ is integrally closed domain. This means that with respect to its prime ideal $p$, localization $\mathbb{Z}_p$ is also integrally closed in its field of fractions. Suppose ...