Rocky Mountain Journal of Mathematics

$\tau $-Regular factorization in commutative rings with zero-divisors

Christopher Park Mooney

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Abstract

Recently there has been a flurry of research on generalized factorization techniques in both integral domains and rings with zero-divisors, namely, $\tau $-factorization. There are several ways that authors have studied factorization in rings with zero-divisors. This paper focuses on the method of regular factorizations introduced by Anderson and Valdes-Leon. We investigate how one can extend the notion of $\tau $-factorization to commutative rings with zero-divisors by using the regular factorization approach. The study of regular factorization is particularly effective because the distinct notions of associate and irreducible coincide for regular elements. We also note that the popular U-factorization developed by Fletcher also coincides since every regular divisor is essential. This will greatly simplify many of the cumbersome finite factorization definitions that exist in the literature when studying factorization in rings with zero-divisors.

Article information

Source
Rocky Mountain J. Math., Volume 46, Number 4 (2016), 1309-1349.

Dates
First available in Project Euclid: 19 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1476884585

Digital Object Identifier
doi:10.1216/RMJ-2016-46-4-1309

Mathematical Reviews number (MathSciNet)
MR3563184

Zentralblatt MATH identifier
1364.13003

Subjects
Primary: 13A05: Divisibility; factorizations [See also 13F15] 13E99: None of the above, but in this section 13F15: Rings defined by factorization properties (e.g., atomic, factorial, half- factorial) [See also 13A05, 14M05]

Keywords
Factorization zero-divisors commutative rings regular and U-factorization

Citation

Mooney, Christopher Park. $\tau $-Regular factorization in commutative rings with zero-divisors. Rocky Mountain J. Math. 46 (2016), no. 4, 1309--1349. doi:10.1216/RMJ-2016-46-4-1309. https://projecteuclid.org/euclid.rmjm/1476884585


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