On power invariant rings

Abstract

Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be an indeterminate and [Formula: see text] be the set of elements [Formula: see text] of [Formula: see text] such that there exists an [Formula: see text]-homomorphism of rings [Formula: see text] with [Formula: see text]. O’Malley called [Formula: see text] to be power invariant (respectively, strongly power invariant) if whenever [Formula: see text] is a ring such that [Formula: see text] is isomorphic to [Formula: see text] (respectively, whenever [Formula: see text] is a ring and [Formula: see text] is an isomorphism of [Formula: see text] onto [Formula: see text]), then [Formula: see text] and [Formula: see text] are isomorphic (respectively, then there exists an [Formula: see text]-automorphism [Formula: see text] of [Formula: see text] such that [Formula: see text]) [M. O’Malley, Isomorphic power series rings, Pacific J. Math. 41(2) (1972) 503–512]. We prove that a ring [Formula: see text] is power invariant in each of the following case: [Formula: see text] [Formula: see text] is a domain in which [Formula: see text] is comparable to each radical ideal of [Formula: see text] (for instance a domain with Krull dimension one), [Formula: see text] [Formula: see text] is a domain in which Jac[Formula: see text] (i.e. the Jacobson radical of [Formula: see text]) is comparable to each radical ideal of [Formula: see text] and [Formula: see text] [Formula: see text] is a Prüfer domain. Also in each of the aforementioned case, we prove that either [Formula: see text] is strongly power invariant or [Formula: see text] is isomorphic to a quasi-local power series ring. Let [Formula: see text] be a unital module over [Formula: see text]. We show that if [Formula: see text] is reduced and strongly power invariant, then Nagata’s idealization ring [Formula: see text] is strongly power invariant (but the converse is false). Ishibashi called a ring [Formula: see text] to be strongly[Formula: see text]-power invariant if whenever [Formula: see text] is a ring and [Formula: see text] is an isomorphism of [Formula: see text] onto [Formula: see text], then there exists an [Formula: see text]-automorphism [Formula: see text] of [Formula: see text] such that [Formula: see text] for each [Formula: see text]. We prove that if [Formula: see text] is a ring in which [Formula: see text] is nil, then [Formula: see text] is strongly[Formula: see text]-power invariant for all positive integer [Formula: see text]. We deduce that every polynomial ring in finitely many indeterminates is strongly[Formula: see text]-power invariant for all positive integer [Formula: see text].