Annihilating Content in Polynomial and Power Series Rings

J. Korean Math. Soc. Published online 2019 Jan 31

Emad Abuosba and Manal Ghanem
The University of Jordan

Abstract : Let R be a commutative ring with unity. If f(x) is a zero-divisor polynomial such that f(x)=c_{f}f₁(x) with c_{f}∈R and f₁(x) is not zero-divisor, then c_{f} is called an annihilating content for f(x). In this case Ann(f)=Ann(c_{f}). We defined EM-rings to be rings with every zero-divisor polynomial having annihilating content. We showed that the class of EM-rings includes integral domains, principal ideal rings, and PP-rings, while it is included in Armendariz rings, and rings having a.c. condition. Some properties of EM-rings are studied and the zero-divisor graphs Γ(R) and Γ(R[x]) are related if R was an EM-ring. Some properties of annihilating contents for polynomials are extended to formal power series rings.

Keywords : Polynomial ring; Power series ring; Annihilating content; EM-ring; Generalized morphic ring; Zero-divisor graph.