Strongly Hopfian subrings of power series rings

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.