Insert row only if it doesn't exist already in MySQL

Insert row only if it doesn't exist already in MySQL

I have a table table with two columns (idA and idB). The table assigns Bs
to As, like this:
A | B
1 | 4
3 | 2
3 | 4
4 | 1
4 | 3 ...
So one A can have multiple Bs and thus shows up in more than one row.
Hence, the table cannot have a primary key and I cannot use a unique
column.
Is there a way to insert new rows only if an equal value pairing does not
already exist, all in one query?
I tried REPLACE INTO and INSERT IGNORE INTO as mentioned here, but both
seem to work for tables with primary keys only.