Adams has a 2.18 ERA in 42 games.
The Padres recalled outfielder Luis Durango to take Adams' place on the 25-man roster.
Adams suffered the injury after delivering a pitch in the Padres' final game before the All-Star break, on Sunday in Denver. He immediately grabbed his left side after throwing a pitch to Ian Stewart. Adams then left the game.
Adams was examined on Thursday afternoon prior to a workout at PETCO Park.
Durango, who has appeared in six games this season with the Padres, getting five hits in 13 at-bats, was hitting .292 with Triple-A Portland with 28 stolen bases.
The Padres also placed right-handed pitcher Mat Latos on the DL on Friday. Latos suffered an injury to his left side, possibly an oblique as well, attempting to hold back a sneeze.
Reliever Ernesto Frieri was added to the roster on Friday. Frieri had a 1.43 ERA and 17 saves with Portland and could end up pitching important innings right away.