The main purpose of this paper is to study the transfer of the strongly Hopfian property to subrings of power series rings. Let A be a commutative ring with identity element and I an arbitrary ideal of A. Among other results, we show that the ring Formula: see text is strongly Hopfian, where Formula: see text is an integer, if and only if the ring A is strongly Hopfian. In the case when Formula: see text, where Formula: see text are different prime numbers, we show that Formula: see text (respectively Formula: see text, where Formula: see text is a nonzero increasing function) is a strongly Hopfian bounded ring if and only if so is A. On the other hand, we construct a large class of non-reduced non-Noetherian rings A (with an arbitrary characteristic) such that the rings Formula: see text and Formula: see text are strongly Hopfian. Many example are provided.
Dabbabi et al. (Wed,) studied this question.