Noun
(algebra, ring theory) Any (two-sided) ideal
I
{\displaystyle I}
such that for arbitrary ideals
P
{\displaystyle P}
and
Q
{\displaystyle Q}
,
P
Q
⊆
I
⟹
P
⊆
I
{\displaystyle PQ\subseteq I\implies P\subseteq I}
or
Q
⊆
I
{\displaystyle Q\subseteq I}
.
In a commutative ring, a (two-sided) ideal
I
{\displaystyle I}
such that for arbitrary ring elements
a
{\displaystyle a}
and
b
{\displaystyle b}
,
a
b
∈
I
⟹
a
∈
I
{\displaystyle ab\in I\implies a\in I}
or
b
∈
I
{\displaystyle b\in I}
.
An ideal P satisfying the commutative definition of prime is sometimes called a completely prime ideal to distinguish it from other merely prime ideals in the ring. Source: Internet
If p is a nonzero non-unit, we say that p is a prime element if, whenever p divides a product ab, then p divides a or p divides b. Equivalently, an element p is prime if and only if the principal ideal (p) is a nonzero prime ideal. Source: Internet
In a principal ideal domain every nonzero prime ideal is maximal, but this is not true in general. Source: Internet
Equivalent conditions for a ring to be a UFD A Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal (a proof is given below). Source: Internet
For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated. Source: Internet
In a Noetherian ring, every prime ideal has finite height. Source: Internet