Abstract Let G be a group, let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>A</m:mi> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mo largeop="true" mathsize="160%" stretchy="false" symmetric="true">⊕</m:mo> <m:mrow> <m:mi>g</m:mi> <m:mo>∈</m:mo> <m:mi>G</m:mi> </m:mrow> </m:msub> <m:msub> <m:mi>A</m:mi> <m:mi>g</m:mi> </m:msub> </m:mrow> </m:mrow> </m:math> {A=\bigoplus_{g\in G}A_{g}} be an epsilon-strongly graded ring over G , let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>R</m:mi> <m:mo>:=</m:mo> <m:msub> <m:mi>A</m:mi> <m:mn>1</m:mn> </m:msub> </m:mrow> </m:math> {R:=A_{1}} be the homogeneous component associated with the identity of G , and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝙿𝚒𝚌𝚂</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathtt{PicS}(R)} be the Picard semigroup of R . In the first part of this paper, we prove that the isomorphism class <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:msub> <m:mi>A</m:mi> <m:mi>g</m:mi> </m:msub> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:math> {[A_{g}]} is an element of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝙿𝚒𝚌𝚂</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathtt{PicS}(R)} for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>g</m:mi> <m:mo>∈</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> {g\in G} . Moreover, the association <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>g</m:mi> <m:mo>↦</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:msub> <m:mi>A</m:mi> <m:mi>g</m:mi> </m:msub> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:math> {g\mapsto[A_{g}]} determines a partial representation of G on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝙿𝚒𝚌𝚂</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathtt{PicS}(R)} which induces a partial action γ of G on the center <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Z</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {Z(R)} of R . Sufficient conditions for A to be an Azumaya <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>R</m:mi> <m:mi>γ</m:mi> </m:msup> </m:math> {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: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>B</m:mi> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="normal">M</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>A</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {B=\operatorname{M}_{n}(A)} is the ring of matrices with entries in A , or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>B</m:mi> <m:mo>=</m:mo> <m:msub> <m:mi>END</m:mi> <m:mi>A</m:mi> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msub> <m:mo largeop="true" mathsize="160%" stretchy="false" symmetric="true">⊕</m:mo> <m:mrow> <m:mi>l</m:mi> <m:mo>∈</m:mo> <m:mi>G</m:mi> </m:mrow> </m:msub> <m:msub> <m:mi>Mor</m:mi> <m:mi>A</m:mi> </m:msub> <m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo>,</m:mo> <m:mi>M</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>l</m:mi> </m:msub> </m:mrow> </m:math> {B=\operatorname{END}_{A}(M)=\bigoplus_{l\in G}\operatorname{Mor}_{A}(M,M)_{l}} is the direct sum of graded endomorphisms of graded left A -modules M with degree l , or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>B</m:mi> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>END</m:mi> <m:mi>A</m:mi> </m:msub> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {B=\operatorname{END}_{A}(M)} where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>M</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>A</m:mi> <m:msub> <m:mo>⊗</m:mo> <m:mi>R</m:mi> </m:msub> <m:mi>N</m:mi> </m:mrow> </m:mrow> </m:math> {M=A\otimes_{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.