UNIFORM AND COUNIFORM DIMENSION OF GENERALIZED INVERSE POLYNOMIAL MODULES

Abstract

Let M be a right R-module, (S, <TEX>${\leq}$</TEX>) a strictly totally ordered monoid which is also artinian and <TEX>${\omega}:S{\rightarrow}Aut(R)$</TEX> a monoid homomorphism, and let <TEX>$[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$</TEX> denote the generalized inverse polynomial module over the skew generalized power series ring [[<TEX>$R^{S,{\leq}},{\omega}$</TEX>]]. In this paper, we prove that <TEX>$[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$</TEX> has the same uniform dimension as its coefficient module <TEX>$M_R$</TEX>, and that if, in addition, R is a right perfect ring and S is a chain monoid, then <TEX>$[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$</TEX> has the same couniform dimension as its coefficient module <TEX>$M_R$</TEX>.