Ok so I am not a math major and i haven't taken an abstract algebra class but i am curoius about the subject. I have been watching video lectures at UCCS at http://cmes.uccs.edu/Fall2007/Math414/archive.php?type=valid and the proffessor talks about groups and rings. In the introduction the groups he states that the definitions of a group are sufficient for solving a linear equation. He states that this is the intuition beheind the idea of a group. Now when he introduces a ring he doesn't seem to give much motivation beheind it. My questions is this: Why are rings important? Is there any intuition behind them that warrent studying them and/or giving them there own name?