This well-organized, readable reference/text provides for the first time a concise introduction to general and multiplicative ideal theory, valid for commutative rings and monoids and presented in the language of ideal systems on (commutative) monoids. Written by a leading expert in the subject, Ideal Systems is a valuable reference for research mathematicians, algebraists and number theorists, and ideal and commutative ring theorists, and a powerful text for graduate students in these disciplines.