in this paper, we show that every element of a discrete module is a sum of two units if and only if its endomorphism ring has no factor ring isomorphic to $z_{2}$. we also characterize unit sum number equal to two for the endomorphism ring of quasi-discrete modules with finite exchange property.

This work explores development in the perception of Japanese singleton and geminate consonant contrasts among native speakers of English learning Japanese as a second language. The primary goal of this paper is to show that the second language (L2) acquisition of phoneme contrasts that are not present in the first language (L1) exhibits development that is predictable from the acoustic properti...