Ash (1998) includes the following areas in his definition of abstract algebra: logic and foundations, counting, elementary number theory, informal set theory, linear algebra, and the theory of linear operators.
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Spring and fall semesters are 16 weeks. They consist of 15 weeks of instruction and a week for finals. Summer sessions tend to vary a lot more from school to school. Individual colleges have a lot more freedom to set summer calendars. I'm on the quarter system, but this is what I have gathered through my research.