Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. There is variety of nice results about approximately Cohen-Macaulay rings. These were done by Goto. In this paper we prove some these for modules and generalize the concept rings to modules. It seen that when $M$ an module, any proper ideal $I$ have $grade(I,M) \geq \dim_R M -\dim_R M/IM -1$. Specially $R$ itself, obtain interval $gr...

The paper presents a solution to the technical problem posed by Girard after the introduction of Ludics of how to define proof nets with quantifiers that commute with multiplicatives. According to the principles of Ludics, the commuting quantifiers have a “locative” nature, in particular, quantified formulas are not defined modulo variable renaming. The solution is given by defining a new corre...