Definition (prime ring):
A ring
is called a prime ring iff its zero ideal is a prime ideal.
Proposition (characterisation of prime ideals):
Let
be a ring, and let
be an ideal. The following are equivalent:
is a prime ideal of ![{\displaystyle R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
is a prime ring
- Whenever
are left ideals in
, then ![{\displaystyle I\cdot J\subseteq P\Rightarrow (I\subseteq P\vee J\subseteq P)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/985b32b1be631f43dac73ccb8c1e1d73b5b97762)
- Whenever
are right ideals in
, then ![{\displaystyle I\cdot J\subseteq P\Rightarrow (I\subseteq P\vee J\subseteq P)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/985b32b1be631f43dac73ccb8c1e1d73b5b97762)
- Whenever
are such that
, then either
or
.
Proof: We'll prove
, since
follows by symmetry. Suppose first that
is a prime ideal. Let
so that
, the zero ideal of
. Then if
is the projection, consider
,
. Then
(since
is a ring homomorphism), so that without loss of generality
, and hence
. Suppose now that
is a prime ring. Let then
such that
. We use the bar notation
(
being the projection) for
. Then we get that
is zero for all
. Then define the ideals
and
, so that then
, hence without loss of generality
and hence
and thus
. Suppose now that 5. holds, and let
be left ideals such that
. Suppose that there existed
so that
and
. Then still
, a contradiction. Note finally that
is trivial.