For an endomorphism

of a ring R, the endomorphism

is called semicommutative if ab=0 implies

=0 for a

R. A ring R is called

-semicommutative if there exists a semicommutative endomorphism

of R. In this paper, various results of semicommutative rings are extended to

-semicommutative rings. In addition, we introduce the notion of an

-skew power series Armendariz ring which is an extension of Armendariz property in a ring R by considering the polynomials in the skew power series ring

. We show that a number of interesting properties of a ring R transfer to its the skew power series ring

and vice-versa such as the Baer property and the p.p.-property, when R is

-skew power series Armendariz. Several known results relating to

-rigid rings can be obtained as corollaries of our results.