Monomial Ideals and Their Decompositions

This textbook on combinatorial commutative algebra focuses on properties of monomial ideals in polynomial rings and their connections with other areas of mathematics such as combinatorics, electrical engineering, topology, geometry, and homological algebra. Aimed toward advanced undergraduate students and graduate students who have taken a basic course in abstract algebra that includes polynomial rings and ideals, this book serves as a core text for a course in combinatorial commutative algebra or as preparation for more advanced courses in the area. The text contains over 600 exercises to provide readers with a hands-on experience working with the material; the exercises include computations of specific examples and proofs of general results. Readers will receive a firsthand introduction to the computer algebra system Macaulay2 with tutorials and exercises for most sections of the text, preparing them for significant computational work in the area. Connections to non-monomial areas of abstract algebra, electrical engineering, combinatorics and other areas of mathematics are provided which give the reader a sense of how these ideas reach into other areas.

W. Frank Moore is an Associate Professor of Mathematics at Wake Forest University. He earned his PhD from the University of Nebraska-Lincoln, and his research is in the homological algebra of commutative and noncommutative rings.

Mark Rogers is a Professor in the Department of Mathematics at Missouri State University. He earned his PhD from Purdue University, and his area of research is commutative algebra.

Sean Sather-Wagstaff is an Associate Professor in Clemson University’s department of Mathematical Sciences. He earned his PhD from the University of Utah, specializing in homological commutative algebra.