Integer ideals

A nonempty set I I in a ring R R is called an ideal if

(1) it is closed under addition: aI,bIa+bI a\in I, b\in I \Rightarrow a+b \in I

(2) it "swallows up" under multiplication: aR,iIaiI a \in R, i \in I \Rightarrow ai \in I .

A proper ideal is one that is not equal to the entire ring.

A proper ideal is prime if abI ab \in I implies that aI a\in I or bI b \in I .

A proper ideal is maximal if there are no ideals in between it and the entire ring: if J J is an ideal, then IJR I \subseteq J \subseteq R implies I=J I=J or J=R J=R.

How many ideals of Z \mathbb Z are prime but not maximal?

It should help to read the ring theory wiki!

Problem Loading...

Note Loading...

Set Loading...