(m,n)-algebraically compactness and (m,n)-pure injectivity

In this paper, we introduce the notion of $(m,n)$-algebr- aically compact modules as an analogue of algebraically compact modules and then we show that $(m,n)$-algebraically compactness and $(m,n)$-pure injectivity for modules coincide. Moreover, further characterizations of a $(m,n)$-pure injective module over a commutative ring are given.