Summary . In this paper, we discuss the properties that hold in finite dimensional vector spaces and related spaces. Mizar language [1], [2], variables are strictly typed, their type conversion requires a complicated process. Our purpose is to formalize some of preserved transformations, contain complexity transformations into paper. Specifically, show such as algebraic structure, subsets, sequ...

the main objective of this study is to swing krull intersection theorem in primary decomposition of rings and modules to the primary decomposition of soft rings and soft modules. to fulfill this aim several notions like soft prime ideals, soft maximal ideals, soft primary ideals, and soft radical ideals are introduced for a soft ring over a given unitary commutative ring. consequently, the p...