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In this work, we deal with partial (co)actions of multiplier Hopf algebras on not necessarily unital algebras. Our main goal is to construct a Morita context relating the coinvariant algebra $R^{\underline{coA}}$ with a certain subalgebra of the smash product $R\#Â$. Besides that, we present the notion of partial Galois coaction, which is closely related to this Morita context.

The representation of rings on finite dimension vector spaces has been generalized to the representation of rings on modules over a commutative ring. Let S be a commutative ring with unity and M an S-module. A representation of ring R with unity on an S-module M is a ring homomorphism from R to the ring of endomorphisms of M. An S-module associated with a representation of R is called a representation module of R. For any ring homomorphism f : R → S, we define a representation of ring R with unity on M via f, and it is called an f-representation of ring R which is a special case of the representation of ring R on an S-module. This S-module associated with the f-representation of ring R is called an f-representation module of R.

In case S is non-commutative, we give a sufficient condition for the S-module M to be a representation module of R. The category of f-representation modules of ring R is Abelian and Morita equivalent to the category of modules over an R-algebra. Thus, if the category of modules over the R-algebra which is equivalent to the category of f-representation modules of R satisfies the Krull-Schmidt Theorem, then the category of f-representation modules of R also satisfies Krull-Schmidt’s Theorem.