A ring R is a left AIP-ring if the left annihilator of any ideal of R is pure as a left ideal. Equivalently, R is a left AIP-ring if R modulo the left annihilator of any ideal is flat. This class of rings includes both right PP-rings and right p.q.-Baer rings (and hence the biregular rings) and is closed under direct products and forming upper triangular matrix rings. It is shown that, unlike the Baer or right PP conditions, the AIP property is inherited by polynomial extensions and has the advantage that it is a Morita invariant property. We also give a complete characterization of a class of AIP-rings which have a sheaf representation. Connections to related classes of rings are investigated and several examples and counterexamples are included to illustrate and delimit the theory.

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In the present note, we continue the study of skew inverse Laurent series ring [Formula: see text] and skew inverse power series ring [Formula: see text], where [Formula: see text] is a ring equipped with an automorphism [Formula: see text] and an [Formula: see text]-derivation [Formula: see text]. Necessary and sufficient conditions are obtained for [Formula: see text] to satisfy a certain ring property which is among being local, semilocal, semiperfect, semiregular, left quasi-duo, (uniquely) clean, exchange, projective-free and [Formula: see text]-ring, respectively. It is shown here that [Formula: see text] (respectively [Formula: see text]) is a domain satisfying the ascending chain condition (Acc) on principal left (respectively right) ideals if and only if so does [Formula: see text]. Also, we investigate the problem when a skew inverse Laurent series ring [Formula: see text] has the same Goldie rank as the ring [Formula: see text] and is proved that, if [Formula: see text] is a semiprime right Goldie ring, then [Formula: see text] is semiprimitive. Furthermore, we study on the relationship between the simplicity, semiprimeness, quasi-Baerness and Baerness property of a ring [Formula: see text] and these of the skew inverse Laurent series ring. Finally, we consider the problem of determining when [Formula: see text] is nilpotent.

Let [Formula: see text] be a ring, [Formula: see text] a strictly ordered monoid and [Formula: see text] a monoid homomorphism. The skew generalized power series ring [Formula: see text] 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 [Formula: see text]-quasi-Armendariz condition on [Formula: see text], 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 [Formula: see text]-quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. The [Formula: see text] 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.