All Superinterfaces:: AbelianGroup<E>, AbelianMonoid<E>, Group<E>, Magma<E>, Monoid<E>, Semiring<E>, Set<E>

All Known Subinterfaces:: Algebra<E,F>, BanachAlgebra<E,F>, CStarAlgebra<E,F>, DivisionRing<E>, Field<E>, GradedAlgebra<E,F>, LieAlgebra<E,S>, Matrix<E>

All Known Implementing Classes:: CliffordAlgebra, Complex, Complexes, DenseMatrix, DoubleField, GenericMatrix, Integer, Integers, MatrixLieAlgebra, MMapMatrix, NativeRealBig, NativeRealDoubleMatrix, NativeRealFloatMatrix, Octonion, Octonions, PolynomialRing, Quaternion, Quaternions, Rational, Rationals, Real, RealBig, RealDouble, RealDoubleMatrix, RealFloat, Reals, SIMDRealDoubleMatrix, SIMDRealFloatMatrix, SparseMatrix, SquareMatrices, TiledMatrix

public interface Ring<E> extends AbelianGroup<E>, Semiring<E>

A ring is an abelian group with a second binary operation (multiplication).

Rings are fundamental algebraic structures that generalize arithmetic. They appear throughout mathematics, from number theory to algebraic geometry.

Mathematical Definition

A ring (R, +, Ãƒâ€”) consists of:

A set R
An addition operation + making (R, +) an abelian group
A multiplication operation Ãƒâ€” satisfying:
- Closure: Ã¢Ë†â‚¬ a, b Ã¢Ë†Ë† R: a Ãƒâ€” b Ã¢Ë†Ë† R
- Associativity: (a Ãƒâ€” b) Ãƒâ€” c = a Ãƒâ€” (b Ãƒâ€” c)
- Distributivity: a Ãƒâ€” (b + c) = (a Ãƒâ€” b) + (a Ãƒâ€” c) and (a + b) Ãƒâ€” c = (a Ãƒâ€” c) + (b Ãƒâ€” c)

Examples