167 views

### The relation between minimal ideals and zero divisors [duplicate]

How we can prove this Theorem. Let $R$ be a reduced ring. Then $a\in ZD(R)$ (the zero divisors of $R$) if and only if $a\in P$ for some minimal prime ideal $P$.
115 views

### Minimal prime ideals consist of zerodivisors [duplicate]

I don't find the proof for this little demonstration ... Let $P$ be a minimal prime ideal of $A$. Show that $P$ is contained in the set of zero divisors of $A$.
100 views

### Minimal prime ideals are made of zero-divisors [duplicate]

Let $R$ be a commutative ring with unity which is not an integral domain. Let $P$ be any minimal prime ideal of $R$. How can I show that $P⊆Z(R)$, where $Z(R)$ denotes the set of zero-divisors of $R.$
7k views

### Showing the set of zero-divisors is a union of prime ideals

I'm working on an exercise from Atiyah and MacDonald's Commutative Algebra, and have hit a bump on Exercise 14 of Chapter 1. In a ring $A$, let $\Sigma$ be the set of all ideals in which every ...
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 ...
2k views

### In a reduced ring the set of zero divisors equals the union of minimal prime ideals.

If $R$ is a reduced commutative ring with identity, why is the set $Z$ of zero divisors the union of minimal prime ideals? I know that $Z$ is a union of associated primes, and that the intersection ...
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 ...
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 ...
364 views

441 views

### What is a minimal prime ideal of a ring

From Wikipedia: A prime ideal P is said to be a minimal prime ideal over an ideal I if it is minimal among all prime ideals containing I. (Note that we do not exclude I even if it is a prime ideal....
241 views

### Why is a non-zero-divisor a unit in zero dimensional ring? [closed]

Let $R$ be a commutative ring of Krull dimension $0$. Let $x\in R\setminus\{0\}$ such that $x$ is non-zero-divisor. Then, how do I prove that $x$ is a unit?
110 views

### A proof about prime ideals

Assume $R$ is commutative. Prove that if $P$ is a prime ideal of $R$ and $P$ contains no zero-divisors then $R$ is an integral domain. Proof: let $ab \in P$ where $ab \not= 0$. that means $a \in P$ ...
### Why the dimension of $R/(a)$ is $0$?
How do I see the following fact? If $R$ has dimension $1$, and $a$ is a non-zerodivisor and non-unit, then $R/(a)$ has dimension $0$. That is saying if $P_1\supset P_2\supset (a)$ are two prime ...
I have a ring $R$ and a prime ideal $P$ of $S=R[t]$ with $t \in P$. I'm trying to prove that if $\mathrm{ht}(P/tS)$ is finite then $\mathrm{ht}(P) > \mathrm{ht}(P/tS)$. Here $\mathrm{ht}(P)$ ...