An integer ring is a mathematical structure that consists of the set of all integers, denoted as ℤ, along with two operations: addition and multiplication. In this context, the integers include positive numbers, negative numbers, and zero. The integer ring satisfies specific properties, such as closure, associativity, and the existence of an additive identity (zero) and a multiplicative identity (one). In an integer ring, every element can be combined using the two operations, and the results will always remain within the set of integers. Additionally, every integer has an additive inverse, meaning for any integer a, there exists another integer -a such that a + (-a) = 0. However, the integer ring does not have multiplicative inverses for all its elements, distinguishing it from fields.