Skew inverse power series rings over a ring with projective socle. (English) Zbl 1458.16050

Paykan, Kamal

Czech. Math. J. 67, No. 2, 389-395 (2017).

Summary: A ring \(R\) is called a right PS-ring if its socle, \(\text{Soc}(R_R)\), is projective. W. K. Nicholson and J. F. Watters [Proc. Am. Math. Soc. 102, No. 3, 443–450 (1988; Zbl 0657.16015)] have shown that if \(R\) is a right PS-ring, then so are the polynomial ring \(R[x]\) and power series ring \(R[[x]]\). In this paper, it is proved that, under suitable conditions, if \(R\) has a (flat) projective socle, then so does the skew inverse power series ring \(R[[x^{-1};\alpha ,\delta ]]\) and the skew polynomial ring \(R[x;\alpha ,\delta ]\), where \(R\) is an associative ring equipped with an automorphism \(\alpha\) and an \(\alpha\)-derivation \(\delta\). Our results extend and unify many existing results. Examples to illustrate and delimit the theory are provided.

MSC:

16W60	Valuations, completions, formal power series and related constructions (associative rings and algebras)
16W70	Filtered associative rings; filtrational and graded techniques
16S36	Ordinary and skew polynomial rings and semigroup rings
16P40	Noetherian rings and modules (associative rings and algebras)

Keywords:

skew inverse power series ring; skew polynomial ring; annihilator; projective socle ring; flat socle ring

Citations:

Zbl 0657.16015