Chain condition on annihilators and strongly Hopfian property in Hurwitz series ring (Q2928447)

Let \(A\) be a commutative ring not necessarily with identity. The Hurwitz series ring \(HA\) is defined by replacing multiplication in the power series ring with the following formula: NEWLINE\[NEWLINE\left(\sum_{n=0}^\infty a_nX^n\right)\left(\sum_{n=0}^\infty b_nX^n\right)=\sum_{n=0}^\infty \left(\sum_{i=0}^n \binom n i a_ib_{n-i}\right)X^n.NEWLINE\]NEWLINE The author relates properties of the ring \(A\) to properties of \(HA\) and of \(hA\) (the ring of Hurwitz polynomials). For example, he proves that if \(A\) is a nil ring of bounded index, then also the ring \(HA\) is nil of bounded index (actually, this statement is formulated in the paper for ideals of a ring \(A\) with identity). The author studies the SFT property of \(HA\) for a nil ring \(A\), the McCoy property for Hurwitz series rings, the chain condition on annihilators \(cc\perp\) in the ring of Hurwitz polynomials, etc. Among other results, the author proves that if the ring \(A\) contains an uncountable field of zero characteristic satisfying \(cc\perp\), then \(hA\) also satisfies \(cc\perp\). Actually, this theorem follows from the Camillo-Guralnick Theorem on \(cc\perp\) for polynomial rings. Indeed, by the proof of Lemma 4.2 in the paper, for any ring \(A\) containing a field of zero characteristic, the function \(f: HA \to A[[X]]\) defined by the formula \(f(\sum_{n=0}^\infty a_nX^n)=\sum_{n=0}^\infty \frac{a_n}{n!}X^n\) is an isomorphism of \(A\)-algebras mapping \(hA\) onto \(A[X]]\) .