Unique factorization in polynomial rings with zero divisors

Abstract

<jats:p> Given a certain factorization property of a ring [Formula: see text], we can ask if this property extends to the polynomial ring over [Formula: see text] or vice versa. For example, it is well known that [Formula: see text] is a unique factorization domain if and only if [Formula: see text] is a unique factorization domain. If [Formula: see text] is not a domain, this is no longer true. In this paper, we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization. </jats:p>

Conjectures for Invertible Diophantine Equations Of 3-Cyclic and 4-Cyclic Refined Integers

“…In the theory of rings, it was very useful to build a bigger ring contains another one, which it is called a ring extension [1][2][11][12][13][14][15]. Recently, it is studied for dealing the four way data analysis using Turiyam ring [16] and its extensive properties [17][18][19] for solving various decision making problems.…”