$\tau$-Atomicity and Quotients of Size Four


  • Richard Erwin Hasenauer Northeastern State University
  • Bethany Kubik University of Minnesota Duluth




Factorization, Commutative Algebra


Given a ring $R$, an ideal $I$ of $R$,  and an element $a\in I$,  we say $a=\lambda b_1\cdots b_k$ is a $\tau_I$-factorization of $a$ if $\lambda$ is any unit and $b_1\equiv\cdots\equiv b_k\pmod{I}$.  In this paper, we investigate the $\tau_I$-atomicity of PIDs with ideals where $R/I$ has size four.

Author Biography

Richard Erwin Hasenauer, Northeastern State University

Associate Professor of Mathematics


D. D. Anderson, and A. M. Frazier, On a general theory of factorization in integral domains, Rocky Mountain J. Math., Vol. 41, No. 3, (2011), 663–705.

D. D. Anderson, and R. M. Ortiz-Albino, Three frameworks for a general theory of factorization, Arab. J. Math. (Springer), 1, No. 1 (2012), 1–16.

D. D. Anderson and J. Reinkoester, Generalized relative primeness in integral domains, Rev. Roumaine Math. Pures Appl., 56, No. 2, (2011), 85–103.

A. A. Florescu, Reduced tau(n) factorizations in Z and tau(n)-factorizations in N, Ph D Thesis, University of Iowa, (2013).

S. M. Hamon, Some topics in tau-factorizations, PhD Thesis, University of Iowa, (2007).

J. R. Juett, Generalized comaximal factorization of ideals, J. Algebra, 512 (2012), 141–166.

J. R. Juett, Some topics in abstract factorization, Thesis (Ph.D.), The University of Iowa, (2013).

A. Mahlum, and C. P. Mooney, Generalized factorization in Z/mZ, Involve. 9, No.3, (2016), 379–393.

C. P. Mooney, Generalized factorization in commutative rings with zero-divisors, Thesis (Ph.D.), The University of Iowa, (2013).

C. P. Mooney, τ-regular factorization in commutative rings with zero-divisors, Rocky Mountain J. Math., 46, No. 4, (2016), 1309–1349.

D. Singmaster, and D. M. Bloom, Problems and Solutions: Solutions of Elementary Problems: E1648, Amer. Math. Monthly, 71, No. 8, (1964), 918–920.




How to Cite

Hasenauer, R. E., & Kubik, B. (2021). $\tau$-Atomicity and Quotients of Size Four. Tamkang Journal of Mathematics, 52(2), 221–228. https://doi.org/10.5556/j.tkjm.52.2021.3241