My niece and son (who are the same age) recently spent a week together in the same house during a family vacation. It was a bit of a social experiment, as they are both incredibly smart but have wildly different personalities.
In the end, they did better than any of us could have expected: They played together, and then they would disagree and separate for a bit before playing together again. For some arguements, there was parental intervention; on others, we let them work it out for themselves.
One thing I was happy to see was that when they did wrong each other, they usually apologized unprompted.
Teaching your child to apologize is a mixed bag. When children are little, they may not understand the meaning of the words, but eventually (you hope) they associate them with wrongdoings and as a means to mend a relationship with someone.
For parents who believe that a prompted apology is an empty gesture, there is some growing research that indicates even a forced apology can make someone feel a little better.
For me, I am a firm believer that practice makes perfect, and that it is a splendid idea to encourage apologies from an early age. One day, they will get it.
Do you prompt your children to say "I'm sorry," even if they don't always mean it? Tell me in the comments.