We extend a lemma by Matsuda about the irreducibility of the binomial X π − 1 in the semigroup ring F [X; G], where F is a field, G is an abelian torsion-free group and π is an element of G of height (0, 0, 0, . . . ).In our extension, G is replaced by any submonoid of (Q + , +). The field F , however, has to be of characteristic 0. We give an application of our main result.
In 2008 N. Q. Chinh and P. H. Nam characterized principal ideal domains as integral domains that satisfy the following two conditions: (i) they are unique factorization domains, and (ii) all maximal ideals in them are principal. We improve their result by giving a characterization in which each of these two conditions is weakened. At the same time we improve a theorem by P. M. Cohn which characterizes principal ideal domains as atomic Bézout domains. We will also show that every PC domain is AP and that the notion of PC domains is incomparable with the notion of pre-Schreier domains (hence with the notions of Schreier and GCD domains as well).2010 Mathematics Subject Classification. Primary 13F15; Secondary 13A05, 13F10.
We give intrinsic characterizations of neural rings and homomorphisms between them. Also we introduce the notion of a basic monomial code map and characterize monomial code maps as compositions of basic monomial code maps. Finally, we characterize monomial isomorphisms between neural codes. Our work is based on the 2015 paper by C. Curto and N. Youngs about neural ring homomorphisms and maps between neural codes and on the 2018 paper by R. Amzi Jeffs about morphisms of neural rings.
The neural rings and ideals as an algebraic tool for analyzing the intrinsic structure of neural codes were introduced by C. Curto, V. Itskov, A. Veliz-Cuba, and N. Youngs in 2013. Since then they were investigated in several papers, including the 2017 paper by S. Güntürkün, J. Jeffries, and J. Sun, in which the notion of polarization of neural ideals was introduced. In our paper we extend their ideas by introducing the notions of polarization of motifs and neural codes. We show that the notions that we introduce have very nice properties which allow the studying of the intrinsic structure of neural codes of length n via the square-free monomial ideals in 2n variables and interpreting the results back in the original neural code ambient space. In the last section of the paper we introduce the notions of inactive neurons, partial neural codes, and partial motifs, as well as the notions of polarization of these codes and motifs. We use these notions to give a new proof of a theorem from the paper by Güntürkün, Jeffries, and Sun that we mentioned above.
Without his support, particularly during qualifying exams, I could never have entered candidacy to begin with, and his genuine interest in students' success does not go unnoticed or unappreciated. I wish to extend my appreciation to the faculties of both the University of Louisville and Indiana University Southeast for providing the preparation and support of my studies. Thanks to April Robinson for initiating my graduate study, to Melissa Neil for inspiring me to teach mathematics, and to Dr. Jessica Taylor for encouraging me to persevere. Thanks to my fellow graduate students for all the understanding and commiserating these last five years. My deepest thanks and gratitude to Dr. Ryann Cartor and Dr. Trevor Leach, for everything. I wish to thank my friends and family, who always believed in me and cheered me on, especially my husband and my parents, for bearing all the difficulties with patience, understanding, grace, and love. Lastly, and above all else, thanks to the Lord Jesus Christ, without whom none of this matters.
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