Give an example of 1) trivial subring & 2) non trivial subring whose all subrings are ideals

1) In any trivial ring, whether it is zero or nonzero ring, all the subrings are ideals. Example : (Q,+,*) ,where * denotes the trivial product on Q

2) The ring of integers is an example of non trivial ring whose all subrings are ideals of Z