Quotient rings

Reducing ring elements modulo an ideal gives a well-defined quotient ring.

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Integral domains
An integral domain is a nonzero commutative ring with identity and no nonzero zero divisors.
Polynomial rings
Polynomials over a ring form a ring with coefficientwise addition and polynomial multiplication.
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.