R-automorphisms ofR[t][[X]]

Joong Kim
1972 Pacific Journal of Mathematics  
Let R be a commutative ring with identity and let R[t] be the polynomial ring in a commuting indeterminate t over R. Let R((")) and R[t]((")) denote the formal power series rings R [ [X "..., X,]] and R [t] [ [Xl ,..., X,]] in n commuting indeterminates X, ,..., X" over R and R [t], respectively. Several papers [3,8,9, lo] have dealt with the structures of the R-endomorphisms (R-automorphisms) of R ((I)) = R [ [Xl], that is, endomorphisms of R"'" that induce the identity mapping on R. Recently
more » ... ilmer and O'Malley [4] have considered the R-endomorphisms of R(("') and determined necessary and sufficient conditions for existence of an R-automorphism of R((")) sending Xi onto pi for i= I,..., n, where each pi is an element of R((")). In fact, the main results of [lo] were generalized by [4] . In this paper we consider Rendomorphism Q of R [t]""" such that $(t) is not necessarily in R[t], and for given elements a, p, ,..., p, of R[t]"")j we show the necessary and sufficient conditions in order that there exists an R-automorphism of R[t]""" sending t onto a and Xi onto pi for each i = I,..., it. For the case n = 1, this author has dealt with this topic in [6] . All rings considered in this paper are assumed to be commutative and contain identity. Throughout this paper the symbols w and w" are used to denote the set of positive and nonnegative integers, respectively.
doi:10.2140/pjm.1972.42.81 fatcat:dxjb7pccjbcnrezwi7hh5nnhja