SSH login without password

If you want to use unix and OpenSSH to automize your tasks. Therefore you need an automatic login from host A / user a to Host B / user b. You don't want to enter any passwords, because you want to call ssh from a within a shell script. First log in on A as user a and generate a pair of authentication keys. Do not enter a passphrase:

a@A:~> ssh-keygen -t rsa
Generating public/private rsa