# Monomial Ideals and Their Decompositions

## Benefits

• Includes tutorials and exercises for Macaulay 2

• Provides hands-on experience with over 600 exercises

• Broadens understanding of monomial ideals in polynomial rings

Textbook

Part of the Universitext book series (UTX)

1. Front Matter
Pages i-xxiv
2. ### Monomial Ideals

1. Front Matter
Pages 1-3
2. W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff
Pages 5-32
3. W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff
Pages 33-79
4. W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff
Pages 81-109
3. ### Monomial Ideals and Other Areas

1. Front Matter
Pages 111-112
2. W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff
Pages 115-159
3. W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff
Pages 161-216
4. ### Decomposing Monomial Ideals

1. Front Matter
Pages 217-218
2. W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff
Pages 221-260
3. W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff
Pages 261-292
5. ### Commutative Algebra and Macaulay2

1. Front Matter
Pages 293-294
2. W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff
Pages 297-329
3. W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff
Pages 331-347
6. Back Matter
Pages 349-387

### Introduction

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.

### Keywords

Macaulay 2 combinatorial commutative algebra irreducible decompositions monomial ideals polynomial rings simplicial complexes modifying monomial ideals decompositions of monomial ideals vertex covers edge ideal construction of Villarreal m-irreducible decompositions parametric decompositions algorithms commutative algebra Dickson’s Lemma Stanley-Reisner ideals Phasor Measurement Units Cohen-Macaulayness Hilbert functions

#### Authors and affiliations

1. 1.Department of MathematicsWake Forest UniversityWinston-SalemUSA
2. 2.Department of MathematicsMissouri State UniversitySpringfieldUSA
3. 3.School of Mathematical and Statistical SciencesClemson UniversityClemsonUSA

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.

### Bibliographic information

• Book Title Monomial Ideals and Their Decompositions
• Authors W. Frank Moore
Mark Rogers
Sean Sather-Wagstaff
• Series Title Universitext
• Series Abbreviated Title Universitext
• DOI https://doi.org/10.1007/978-3-319-96876-6
• Copyright Information Springer Nature Switzerland AG 2018
• Publisher Name Springer, Cham
• eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
• Softcover ISBN 978-3-319-96874-2
• eBook ISBN 978-3-319-96876-6
• Series ISSN 0172-5939
• Series E-ISSN 2191-6675
• Edition Number 1
• Number of Pages XXIV, 387
• Number of Illustrations 55 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site

## Reviews

“Primarily directed at advanced undergraduates, the text is also a valuable resource for graduate students and researchers who wish to learn more about the subject, providing an introduction to active research topics in combinatorial commutative algebra and its applications. … the authors' presentation of monomial decompositions and their applications is an exciting, enlightening read and will serve an individual reader or class instructor well.” (Timothy B. P. Clark, Mathematical Reviews, October, 2019)

“Each definition includes examples of reasonably common structures … . This style makes the text accessible to advanced undergraduates. … it will be useful to those who work in symbolic computation and theory.” (Paul Cull, Computing Reviews, May 13, 2019)