Linked Questions

41
votes
3answers
6k views

Why does a minimal prime ideal consist of zerodivisors?

Let $A$ be a commutative ring. Suppose $P \subset A$ is a minimal prime ideal. Then it is a theorem that $P$ consists of zero-divisors. This can be proved using localization, when $A$ is noetherian: $...
12
votes
2answers
3k views

An ideal that is maximal among non-finitely generated ideals is prime.

I've been doing some old exam problems and I've come across a problem that I've answered, but my gut is telling me that there's something I'm glossing over. Let $R$ be a commutative ring with ...
18
votes
3answers
4k views

Complement of maximal multiplicative set is a prime ideal

Let $R$ be a commutative ring with identity. I've been trying to prove the following: If $S \subset R$ is a maximal multiplicative set, then $R \setminus S$ is a prime ideal of $R$. I have spent ...
8
votes
2answers
2k views

$x$ not nilpotent implies that there is a prime ideal not containing $x$.

Let $\mathscr{N}(R)$ denote the set of all nilpotent elements in a ring $R$. I have actually done an exercise which states that if $x \in \mathscr{N}(R)$, then $x$ is contained in every prime ideal ...
4
votes
2answers
2k views

In a noetherian integral domain every non invertible element is a product of irreducible elements

I want to prove that in a noetherian ring $R$ which is also an integral domain, every non invertible element can be expressed as product of irreducible elements. I really do not know where to start. ...
7
votes
2answers
1k views

The set consisting of all zero divisors in a commutative ring with unity contains at least one prime ideal

I'm asked to prove that the set consisting of all zero divisors in a commutative ring with unity contains at least one prime ideal. I can't even start in the proof, I've just defined my set but cant ...
2
votes
2answers
2k views

Power of prime ideal

I am beginner in algebra. I want to know if every power of a prime ideal is a principal ideal. Is the statement correct or is there a counterexample?
7
votes
2answers
604 views

Is ideal an “anti-field”?

I am comparing theorems on normal subgroup and ideal from Fraleigh's, and come to this strange intuition. I hope my conclusion does not screw up, I hope I won't get ridiculed: Theorem 15.18: $M$is ...
5
votes
3answers
4k views

Not a Zero Divisor

Let $R$ be a commutative ring. Then we say $a \in R$ is a zero divisor if there exists $b \neq 0$ such that $ab = 0$. I want to know what it means to not be a zero divisor. So I tried to negate the ...
3
votes
2answers
408 views

Extending morphisms with Zorn's lemma

I have stumbled upon a remarkable similarity between the proof of Baer's criterion and an extension theorem in field theory. Here are the statements: Baer's criterion: Let $R$ be a ring. A left $R$-...
1
vote
1answer
429 views

A question on irreducible elements in principal ideal domains.

In my book it says that if $d$ is an element in a principal ideal domain $D$ that cannot be written as a product of irreducible elements then it follows that $d$ is not irreducible. I dont understand ...
1
vote
2answers
206 views

Is $Z(R)$ a maximal ideal?

If $p$ and $q$ are two maximal ideals in the set of zero-divisors in a ring $R$ with non-zero intersection between $p$ and $q$. does the set of all zero-divisors are a maximal ideal and equal the ...
1
vote
1answer
117 views

Any deeper “duality” between non-zero-divisors and units of a ring?

I'm reading Aluffi's algebra book at the moment -- specifically, I'm on the introductory rings/modules chapter. I noticed two interesting pieces of information: in a (not necessarily commutative) ...
1
vote
0answers
32 views

What is the intuitive reason to expect maximality condition giving primeness of ideals in general?

Here I will quote a few basic examples of maximality condition yielding primeness of ideals, though I knew how to prove them and knew the statement. I do not think I have a good intuitive answer to ...