Special Properties of Rings of Generalized Power Series

Abstract

Let R be a ring and (S, ≤) a strictly ordered monoid. Properties of the ring [[R ^S,≤]] of generalized power series with coefficients in R and exponents in S are considered in this paper. It is shown that [[R ^S,≤]] is reduced (2-primal, Dedekind finite, clean, uniquely clean) if and only if R is reduced (2-primal, Dedekind finite, clean, uniquely clean, respectively) under some additional conditions. Also a necessary and sufficient condition is given for rings under which the ring [[R ^S,≤]] is a reduced left PP-ring.

Key Words:

• 2-primal
• Dedekind finite
• PP-ring
• Generalized power series

Acknowledgments

The author wishes to express his sincere thanks to the referee for his/her valuable suggestions. This work was supported by National Natural Science Foundation of China (10171082) and NWNU-KJCXGC212, TRAPOYT.

Notes

^#Communicated by E. Puczylowski.