In this paper, we define soft intersection semigroups, soft intersection left (right, two-sided) ideals and bi-ideals of semigroups, give their properties and interrelations and we characterize regular, intra-regular, completely regular, weakly regular and quasi-regular semigroups in terms of these ideals.

In this paper, we introduce the concept of wsa-supplements and investigate the objects of the class of short exact sequences determined by wsa-supplement submodules, where a submodule U of a module M is called a wsa-supplement in M if there is a submodule V of M with $U+V=M$ and $U\cap V$ is weakly semiartinian. We prove that a module M is weakly semiartinian if and only if every submodule of M is a wsa-supplement in M. We introduce $CC$-rings as a generalization of C-rings and show that a ring is a right $CC$-ring if and only if every singular right module has a crumbling submodule. The class of all short exact sequences determined by wsa-supplement submodules is shown to be a proper class which is both injectively and co-injectively generated. We investigate the homological objects of this proper class along with its relation to $CC$-rings.

In this paper, we provide the basic properties of (semi)simple hypermodules. We show that if a hypermodule M is simple, then $\left(End\right(M),·)$ is a group, where $End\left(M\right)$ is the set of all normal endomorphisms of M. We prove that every simple hypermodule is normal projective with a zero singular subhypermodule. We also show that the class of semisimple hypermodules is closed under internal direct sums, factor hypermodules, and subhypermodules. In particular, we give a characterization of internal direct sums of subhypermodules of a hypermodule.

We introduce the concept of (strongly) $ss$-radical supplemented modules. We prove that if a submodule $N$ of $M$ is strongly $ss$-radical supplemented and $Rad(M/N)=M/N$, then $M$ is strongly $ss$-radical supplemented. For a left good ring $R$, we show that $Rad(R)\subseteq Soc(_{R}R)$ if and only if every left $R$-module is $ss$-radical supplemented. We characterize the rings over which all modules are strongly $ss$-radical supplemented. We also prove that over a left $WV$-ring every supplemented module is $ss$-supplemented.

This study is motivated by the recently published papers on normal injective and normal projective hypermodules. We provide a new characterization of the normal injective and normal projective hypermodules by using the splitting of the short exact sequences of hypermodules. After presenting some of their fundamental properties, we show that if a hypermodule is normal projective, then every exact sequence ending with it is splitting. Moreover, if a hypermodule is normal injective, then every exact sequence starting with it is splitting as well. Finally, we investigate the relationships between semisimple, simple, normal injective, and normal projective hypermodules.

In this paper, we define the concept of pseudo-prime subhypermodules of hypermodules as a generalization of the prime hyperideal of commutative hyperrings. In particular, we examine the spectrum of the Zariski topology, which we built on the element of the pseudo-prime subhypermodules of a class of hypermodules. Moreover, we provide some relevant properties of the hypermodule in this topological hyperspace.