Epsilon-strongly graded rings: Azumaya algebras and partial crossed products

Let G be a group, let A = circle plus(g is an element of G) A(g) be an epsilon-strongly graded ring over G, let R := A(1) be the homogeneous component associated with the identity of G, and let PicS(R) be the Picard semigroup of R. In the first part of this paper, we prove that the isomorphism class [Ag] is an element of PicS(R) for all g is an element of G. Moreover, the association g (sic) [A(g)] determines a partial representation of G on PicS(R) which induces a partial action. of G on the center Z( R) of R. Sufficient conditions for A to be an Azumaya R-gamma-algebra are presented if R is commutative. In the second part, we study when B is a partial crossed product in the following cases: B = M-n(A) is the ring of matrices with entries in A, or B = ENDA(M) = circle plus(l is an element of G) Mor(A)( M, M)(l) is the direct sum of graded endomorphisms of graded left A-modules M with degree l, or B = ENDA(M) where M = A (circle times R) N is the induced module of a left R-module N. Assuming that R is semiperfect, we prove that there exists a subring of A which is an epsilon-strongly graded ring over a subgroup of G and it is graded equivalent to a partial crossed product.