Associative left or right neat-rings $S$ with the descending chain condition (dcc) on right or left (principal) $S$-subgroups have been considered by Laxton [1], Beidleman [2], high [3] and others. We iniciate here a study of (nonassociative) left near-rings $S$ with one or both dec properties ($*$) and ($**$). He obtain results on the existence of the identity of $S$, and the right or left i invertibility of certain elements in $S$, particularly for the case when the additive group of $S$ is simple.