Modules whose special local summands are summands

Let [Formula: see text] be a right [Formula: see text]-module. A summand [Formula: see text] of [Formula: see text] is called special if there exists another summand [Formula: see text] of [Formula: see text] with [Formula: see text] and [Formula: see text]. A local summand [Formula: see text] of [Formula: see text] is called special if each [Formula: see text] is a special summand of [Formula: see text]. In this paper, we investigate modules whose special local summands are summands, obtaining new results and extending and sharpening existing results.