Rivalry (noun)

1. Competition for the same objective or for superiority in the same field.
2. A state of competition or enmity between two people or groups.

Origin:

Early 17th century (in the sense "a state of being rivals"): from rival + -ry.

Examples:

1. The intense rivalry between the two teams.
2. The rivalry between the two companies.
3. The long-standing rivalry between the two families.
4. The rivalry between the two brothers.
5. The fierce rivalry between the two countries.