mathbbZn

The notation mmathbbZn refers to the set of integers modulo n. This set consists of n distinct elements, which can be represented by the integers from 0 to n-1. Arithmetic operations, such as addition and multiplication, are performed on these elements, and the result is then reduced modulo n. This means that if a calculation results in a value greater than or equal to n, n is repeatedly subtracted until the result is within the range [0, n-1]. Similarly, if the result is negative, n is repeatedly added until it falls within this range.

The set mmathbbZn, along with the operations of addition and multiplication modulo n, forms a mathematical structure called a ring. This ring is particularly important in number theory, abstract algebra, and computer science. For example, mmathbbZ2 is the set {0, 1} and is fundamental to binary arithmetic and Boolean algebra. When n is a prime number, mmathbbZn forms a field, denoted as F_p or GF(p), which has applications in cryptography and coding theory. The properties of mmathbbZn, such as its commutativity and associativity under addition and multiplication, make it a versatile tool for various mathematical investigations.