rings.hpp

#ifndef __RINGS_HPP_
#define __RINGS_HPP_

namespace Rings {

class Ring_Element {
public:
  virtual bool is_one() const = 0;
  virtual bool is_zero() const = 0;
  virtual bool has_inverse() const = 0;
  virtual bool is_commutative() const { return false; }
  virtual bool is_cancellable() const { return false; }

  virtual bool operator == (const Ring_Element &) const = 0;
  virtual bool operator != (const Ring_Element &) const = 0;

  virtual const Ring_Element & operator + (const Ring_Element & ) const = 0;
  virtual const Ring_Element & operator - (const Ring_Element & ) const = 0;
  virtual const Ring_Element & operator * (const Ring_Element & ) const = 0;
};

class Commutative_Ring_Element : public Ring_Element {
public:
  virtual bool is_commutative() const override { return true; }
};

class Integral_Domain_Element : virtual public Commutative_Ring_Element {
public:
  virtual bool is_cancellable() const override { return not is_zero(); }
};

class Field_Element : virtual public Integral_Domain_Element {

public:

  virtual bool has_inverse() const override { return not is_zero(); }

  virtual Field_Element & operator / (const Field_Element &) const = 0;
  virtual Field_Element & inverse() const = 0;

};

}

#endif