Inheritance, shared primary keys between parent and child tables.

I thought the primary key was NOT shared between parent/child tables
through inheritance.

On reading the documentation section for inheritance in Postgresql
http://www.postgresql.org/docs/9.1/static/ddl-inherit.html
It seems obvious that the primary key of the parent table is not shared by
the child table(am I miss-understanding?)

My experience from playing around with this indicates otherwise.

For example

CREATE TABLE account (
id SERIAL PRIMARY KEY
);

CREATE TABLE account_admin (
cash INTEGER
)INHERITS (account);

CREATE TABLE account_user (
credit INTEGER
)INHERITS (account);

INSERT INTO account_admin(cash) VALUES(12);
INSERT INTO account_user(credit) VALUES(12);
INSERT INTO account_user(credit) VALUES(12);
INSERT INTO account_admin(cash) VALUES(12);

My understanding of the documentation is:

SELECT id FROM account_admin;

should return 1,2

actually returns: 1,4;

And:

SELECT id FROM account_user;
should return: 1,2
actually returns: 2,3

While
SELECT id FROM account;
returns: 1,2,3,4 as expected

Does this mean that child tables of a parent share the same primary key?