Definition. prime ideal [math-0012]

Lîm Tsú-thuàn

Let AA be a ring and II be an ideal, the followings are equivalent conditions to say that II is prime

  1. II is prime if ab∈Iab \in I than a∈Ia \in I or b∈Ib \in I for all a,b∈Aa,b \in A
  2. II is prime if A/IA / I is an integral domain

Proof [local-2]

Backward [local-0]

Let A/IA / I be an integral domain, that's say if x,y∈A/Ix, y \in A / I and xy=0xy = 0, then x=0x = 0 or y=0y = 0. Let (a+I)(b+I)(a + I)(b + I) be the zero element of II (i.e. (0∈A)+I(0 \in A) + I), then ab+I=Iab + I = I. Hence a+I=Ia + I = I or b+I=Ib + I = I, implies a∈Ia \in I or b∈Ib \in I.

Forward [local-1]

Let II be a prime ideal, let

(a+I)(b+I)=0+I=I(a+I)(b+I)=0+I = I

then ab∈Iab \in I and therefore, a∈Ia \in I or b∈Ib \in I. Hence a+Ia + I or b+Ib + I is the zero coset in A/IA / I.