On ϕ-piecewise Noetherian rings

Abstract

The purpose of this paper is to introduce two new classes of rings that is closely related to the classes of piecewise Noetherian domains and piecewise w-Noetherian domains. Let $\mathcal{H}=\left\{R\right|R$ is a commutative ring and $\text{Nil}(R)$ is a divided prime ideal of $R\}.$ Let $R\in \mathcal{H}$ be a ring with total quotient ring T(R), and set $\varphi :T(R)\to R_{\text{Nil}(R)}$ such that $\varphi \left(\frac{a}{b}\right)=\frac{a}{b}$ for every $a\in R$ and every $b\in R\mathrm{\setminus }Z(R).$ Then $\varphi$ is a ring homomorphism from T(R) into $R_{\text{Nil}(R)},$ and $\varphi$ restricted to R is also a ring homomorphism from R into $R_{\text{Nil}(R)}$ given by $\varphi (x)=\frac{x}{1}$ for every $x\in R.$ At this setting, we introduce the notions of $\varphi$-piecewise Noetherian rings and $\varphi$-piecewise w-Noetherian rings and we show that the theories of $\varphi$-piecewise Noetherian (resp. $\varphi$-piecewise w-Noetherian) rings resemble those of piecewise Noetherian (resp. piecewise w-Noetherian) domains.

Keywords:

• ϕ-piecewise Noetherian rings
• ϕ-piecewise w-Noetherian rings
• trivial ring extension

2010 Mathematics Subject Classification:

• Primary 13B99
• Secondary 13A15
• 13G05
• 13B21

Acknowledgements

The authors would like to express their sincere thanks for the referee for his/her helpful suggestions and comments, which have greatly improved this paper.