On some classes of pure subhypermodules and some classes of pure subacts over monoid

The notion of a pure subhypermodule and so pure subhypermodule relative to subhypermodule are introducing. Some properties of these concepts have been studied. In this work the notion of a \(E_n\)-pure subact and so \(E_n\)-pure subact relative to subact have been introduced. Some properties of these concepts are studing. Prove that X is pure subhypermodule if and only if foreach finite sets \(\{mi\} \in M, \{ni\} \in X\) with \(\{r_{ij}\} \in R\) and \(nj = \sum _{i=1}^{k}{r_{ij}m_i}, j = 1, 2,\ldots , l,\) there is a \(\{ x_i \} \in X\) which is finite set, when \(nj -\sum _{i=1}^{k}{r_{ij}x_i} \in X \cap K\) for each subhypermodule K, and A hypermodule M owns the pure intersection property if and only if \(\left( zN \cap zK\right) =z\left( \ N\cap K \right) \) for each \(z \in R\) and for all pure subhypermodules N, K in M. Also, prove that for act, If M owns the \(E_n\)-pure subact intersection property, then each \(E_n\)-pure subact in M has the \(E_n\)-pure subact intersection property, and Put X is \(E_n\)-pure subact in M. M has \(E_n\)-pure sub-act intersection property, if and only if, \(\frac{M}{X}\) has \(E_n\)-pure subact intersection property.