The Importance of Being Prime ⋆ , a Nontrivial Generalization for Nonunique Factorizations

AbstractThe notion of primeness is the key to the phenomenon of unique factorization. In particular, when unique factorization in a monoid fails, the arithmetic of that monoid is determined by the irreducible elements which are not prime. We illustrate this with examples of easy-to-understand monoids which are, for the most part, multiplicative submonoids of the natural numbers. Through these examples, we examine the ω-invariant, which offers a quantification of both primeness and nonunique factorization. We close by shifting gears and illustrating the same concepts in noncommutative semigroups, again by using relatively simple constructions involving positive integers.MSC: 13A0511A51 Additional informationNotes on contributorsNicholas R. BaethNICHOLAS R. BAETH passed away on December 11, 2021, at the age of 43, after a brief struggle with cancer. He was a professor at the University of Central Missouri for 13 years and then at Franklin & Marshall College for 312 years. Baeth was a member of the MAA for 25 years. His passion was teaching and doing research with undergraduates, and he was a specialist in algebra. Baeth was a member of the 2005 class for Project NExT, and served on the Merten M. Haase Prize committee. More information about his life and work can be found in his department’s memorial statement https://www.fandm.edu/uploads/files/970943588902998302-nick-baeth-memorial-statement.pdf.Scott T. ChapmanSCOTT T. CHAPMAN is a Texas State University System Regents’ Professor and SHSU Distinguished Professor at Sam Houston State University in Huntsville, Texas. He served as Editor of the American Mathematical Monthly during the period 2012–2016. He is currently serving as Editor-in-Chief at Communications in Algebra. His editorial work, numerous publications in the area of non-unique factorizations, and years of directing REU Programs, led to his designation in 2017 as a Fellow of the American Mathematical Society.