Modules

Modules generalize vector spaces by allowing scalars from a ring rather than a field.

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Rings
A ring supports addition and multiplication linked by distributive laws.
Unique factorization domains
Nonzero nonunits factor into irreducibles uniquely up to order and units.
Binary operations
A binary operation combines two elements into one in a specified set.
Cosets and Lagrange's theorem
Cosets partition a finite group, so a subgroup's order divides the group's order.
Cyclic groups
A cyclic group is generated by repeated powers of a single element.