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  • articleNo Access

    Quasi-Armendariz generalized power series rings

    Let R be a ring, (S,) a strictly ordered monoid and ω:SEnd(R) a monoid homomorphism. The skew generalized power series ring R[[S,ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. We initiate the study of the (S,ω)-quasi-Armendariz condition on R, a generalization of the standard quasi-Armendariz condition from polynomials to skew generalized power series. The class of quasi-Armendariz rings includes semiprime rings, Armendariz rings, right (left) p.q.-Baer rings and right (left) PP rings. The (S,ω)-quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. The 2×2 formal upper triangular matrix rings of this class are characterized. We conclude some characterizations for a skew generalized power series ring to be semiprime, quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and left AIP. Examples to illustrate and delimit the theory are provided.

  • articleNo Access

    Skew inverse Laurent series extensions of weakly principally quasi Baer rings

    A ring R is called weakly principally quasi Baer or simply (weakly p.q.-Baer) if the right annihilator of a principal right ideal is left s-unital by left semicentral idempotents, which implies that R modulo the right annihilator of any principal right ideal is flat. We study the relationship between the weakly p.q.-Baer property of a ring R and those of the skew inverse series rings R((x1;σ,δ)) and R[[x1;σ,δ]], for any automorphism σ and σ-derivation δ of R. Examples to illustrate and delimit the theory are provided.