Power series over strongly Hopfian bounded rings

ABSTRACT

Let R be a commutative ring with identity. We show that R[[X]] is strongly Hopfian bounded if and only if R has a strongly Hopfian bounded extension T such that I_c(T) contains a regular element of T. We deduce that if R[[X]] is strongly Hopfian bounded, then so is R[[X,Y]] where X,Y are two indeterminates over R. Also we show that if R is embeddable in a zero-dimensional strongly Hopfian bounded ring, then so is R[[X]] (this generalizes most results of Hizem [11]). For a chained ring R, we show that R[[X]] is strongly Hopfian if and only if R is strongly Hopfian.

KEYWORDS:

• Chained ring
• embeddability in a zero-dimensional ring
• power series ring
• SFT-ring
• strongly Hopfian (bounded) ring

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A15
• 13F25
• 13F30

Acknowledgment

The author is grateful to Professor Sana Hizem for valuable discussions. The author would like to thank the referee for several valuable comments and suggestions.