Proving Integers as an Integral Domain

Prove that the set of all integers with regular addition and multiplication is an integral domain by showing that it satisfies all the conditions except for the existence of a multiplicative inverse.

In this problem, you are exploring the foundational concept of an integral domain within the context of number theory and abstract algebra by analyzing the properties of the set of all integers under regular addition and multiplication. The goal here is to demonstrate, formally, that integers satisfy the critical conditions of an integral domain, aside from the presence of multiplicative inverses for every element. An integral domain is essentially defined as a non-zero commutative ring with no zero divisors.

When tackling this kind of proof, you'll need to engage with several core properties intrinsic to ring theory, such as associativity, commutativity, and distributivity, which are readily verified for the integers. Additionally, verifying the absence of zero divisors is a notable exercise, reflective of the structural characteristics of integers, highlighting that if the product of two integers is zero, then at least one of the integers must be zero itself. This involvement of theoretical constructs and logical deductions to establish that integers naturally serve as an example of an integral domain is akin to addressing a standard proof structure often encountered in higher mathematics.

Overall, this problem teases apart the conceptual threads of integral domains from the broader category of rings. Understanding why integers fall into this category not only reinforces your comprehension of integral domains but also enhances your theoretical insight into the discipline of abstract algebra. This problem encourages you to harness the axiomatic properties of mathematical structures as combinatorial tools essential for constructing formal logical arguments, an invaluable skill set as you deepen your exploration of algebraic systems.