Narrow Results

Let $R$ be a ring, $(S,\le )$ a strictly ordered monoid and $\omega :S\to \mathrm{\text{End}}(R)$ a monoid homomorphism. The skew generalized power series ring $R[[S,\omega ]]$ 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 $(S,\omega )$-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 $(S,\omega )$-quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. The $2\times 2$ 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.

A generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of differential inverse power series type (or simply, $\mathcal{D}\mathcal{I}\mathcal{P}\mathcal{S}$-quasi-Armendariz), is introduced and studied. It is shown that the $\mathcal{D}\mathcal{I}\mathcal{P}\mathcal{S}$-quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. Various classes of non-semiprime $\mathcal{D}\mathcal{I}\mathcal{P}\mathcal{S}$-quasi-Armendariz rings are provided, and a number of properties of this generalization are established. Some characterizations for the differential inverse power series ring $R[[x^{-1};\delta ]]$ to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and left AIP are concluded, where δ is a derivation on the ring R. Finally, miscellaneous examples to illustrate and delimit the theory are given.

For a ring endomorphism $\alpha$, a generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of skew Hurwitz series type (or simply, $\alpha$-${𝒮}{\mathscr{H}}{𝒬}{𝒜}$), is introduced and studied. It is shown that the ${𝒮}{\mathscr{H}}{𝒬}{𝒜}$-rings are closed upper triangular matrix rings, full matrix rings and Morita invariance. Some characterizations for the skew Hurwitz series ring $(HR,\alpha )$ to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and semiprime are concluded.