mathbbZ

mathbbZ denotes the set of integers: {..., -2, -1, 0, 1, 2, ...}. It is a subset of the real numbers and is closed under ordinary addition and multiplication. The symbol is standard in mathematical notation, using double-struck Z to distinguish integers from other number systems.

As a ring, Z is a commutative ring with unity (1) and is an integral domain. It is also a Euclidean domain with respect to the absolute value, hence a principal ideal domain and a unique factorization domain. A key consequence is the Fundamental Theorem of Arithmetic: every nonzero integer factors uniquely up to sign into primes.

As an additive group, Z is infinite cyclic, generated by 1; every element can be written as n·1 for some integer n. The ring Z is a subring of the real numbers and of the rational numbers, via the usual inclusion. It is countably infinite.

Universal property: Z is the initial object in the category of unital commutative rings; for any ring R there is a unique ring homomorphism f: Z → R sending 1 to 1_R. This underlies the construction of Z-algebras and modules and explains why every ring contains a canonical copy of the integers.

Applications and role: Z serves as the prototypical coefficient ring in number theory and algebra. It forms the basis for the theory of integers, prime factorization, and arithmetic, and provides the foundational ring over which modules and many algebraic structures are built.