Although there are many types of ring extensions, simple extensions have yet to be thoroughly explored in one book. Covering an understudied aspect of commutative algebra, Simple Extensions with the Minimum Degree Relations of Integral Domains presents a comprehensive treatment of various simple extensions and their properties. In particular, it examines several properties of simple ring extensions of Noetherian integral domains. As experts who have been studying this field for over a decade, the authors present many arguments that they have developed themselves, mainly exploring anti-integral, super-primitive, and ultra-primitive extensions. Within this framework, they study certain properties, such as flatness, integrality, and unramifiedness. Some of the topics discussed include Sharma polynomials, vanishing points, Noetherian domains, denominator ideals, unit groups, and polynomial rings. Presenting a complete treatment of each topic, Simple Extensions with the Minimum Degree Relations of Integral Domains serves as an ideal resource for graduate students and researchers involved in the area of commutative algebra.

This book constitutes the proceedings of the 14th International Workshop on Knowledge Management and Acquisition for Intelligent Systems, PKAW 2016, held in Phuket, Thailand, in August 2016. The 16 full papers and 5 short papers included in this volume were carefully reviewed and selected from 61 initial submissions. They deal with knowledge acquisition and machine learning; knowledge acquisition and natural language processing; knowledge acquisition from network and big data; and knowledge acquisition and applications.

This book constitutes the proceedings of the 15th International Workshop on Knowledge Management and Acquisition for Intelligent Systems, PKAW 2018, held in Nanjing, China, in August 2018. The 15 full papers and 7 short papers included in this volume were carefully reviewed and selected from 51 initial submissions. They cover the methods and tools as well as the applications related to developing a knowledge base, healthcare, financial systems, and intelligent systems.

Although there are many types of ring extensions, simple extensions have yet to be thoroughly explored in one book. Covering an understudied aspect of commutative algebra, Simple Extensions with the Minimum Degree Relations of Integral Domains presents a comprehensive treatment of various simple extensions and their properties. In particular, it examines several properties of simple ring extensions of Noetherian integral domains. As experts who have been studying this field for over a decade, the authors present many arguments that they have developed themselves, mainly exploring anti-integral, super-primitive, and ultra-primitive extensions. Within this framework, they study certain properties, such as flatness, integrality, and unramifiedness. Some of the topics discussed include Sharma polynomials, vanishing points, Noetherian domains, denominator ideals, unit groups, and polynomial rings. Presenting a complete treatment of each topic, Simple Extensions with the Minimum Degree Relations of Integral Domains serves as an ideal resource for graduate students and researchers involved in the area of commutative algebra.

Obadiah is one of a series of leadership commentaries, which Dr. Clinton and others are writing. Obadiah particularly emphasizes how God faithfully performs His promises to His people. Leaders are people who must know God as The Promise Keeper.