# An exercise in Algebra by Thomas W. Hungerford

• October 17th 2008, 11:26 PM
chipai
An exercise in Algebra by Thomas W. Hungerford
A commutive ring wih unity has a minmal prime ideal contains all zero divisor and all nonunits are zero divisor
prove
all nonunits are nilpotent

this is an exercise in Algebra by Thomas W. Hungford page 148
• October 18th 2008, 01:43 AM
NonCommAlg
Quote:

Originally Posted by chipai
A commutive ring wih unity has a minmal prime ideal which contains all zero divisors and all nonunits are zero divisor. prove that all nonunits are nilpotent.

this is an exercise in Algebra by Thomas W. Hungford page 148 about local ring.

let $R$ be the ring and $P$ the minimal prime. so $R - P$ is exactly the set of units of $R.$ hence any proper ideal of $R$ is contained in $P.$ since $P$ is a minimal prime, it cannot contain any prime ideal

properly. thus $P$ is the unique prime ideal of $R.$ now suppose $x \in R$ is a non-unit. so $x \in P.$ suppose $x$ is not nilpotent. let $\mathcal{S}=\{x^n : \ n = 1, 2, \cdots \}.$ then $0 \notin \mathcal{S},$ because we assumed that $x$ is

not nilpotent. let $\mathcal{C}=\{I \lhd R: \ I \cap \mathcal{S} = \emptyset \},$ which is a non-empty set because $<0> \in \mathcal{C}.$ apply Zorn's lemma to find $Q,$ a maximal element of $\mathcal{C}.$ we claim that $Q$ is prime: suppose $ab \in Q$ but

$a \notin Q, \ b \notin Q.$ then $Q \subset Q + Ra, \ Q \subset Q+Rb.$ therefore $(Q+Ra) \cap \mathcal{S} \neq \emptyset, \ (Q+Rb) \cap \mathcal{S} \neq \emptyset,$ by maximality of $Q.$ so there exists $i,j$ such that $x^i \in Q+Ra, \ x^j \in Q+Rb.$ but then:

$x^{i+j} \in (Q+Ra)(Q+Rb) \subseteq Q+Rab =Q,$ because $ab \in Q.$ thus $Q \cap \mathcal{S} \neq \emptyset,$ which is contradiction! so $Q$ is prime and hence $Q=P.$ thus $P \cap \mathcal{S} = \emptyset,$ which is false because $x \in P \cap \mathcal{S}. \ \ \ \Box$

Remark 1: in general, in a (not even necessarily commutative) ring $R,$ given a multiplicatively closed set $S \subset R$ with $0 \notin S,$ we can always find a prime ideal $Q$ which is contained in $R - S.$

Remark 2: if you already know about the nilradical of a commutative ring, then your problem is trivial: since $P$ is the only prime ideal of $R,$ the nilradical of $R$ is $P.$ but we know that the nilradical

is exactly the set of nilpotent elements of $R.$

Remark 3: the information given in the problem about zero-divisors is only used to conclude that $R-P$ is exactly the set of units of $R.$