Abstract :
It is proved that if M is a uniform module such that M( 1) is CS, then its endomorphism ring is local. As a consequence, it is shown that every 1-Σ-CS module is Σ-CS. This solves open questions asked in [Contemp. Math. 259 (2000) 467, J. Algebra 262 (2003) 194]. On the way to these results, it is also shown that the endomorphism ring of a uniform module M is local provided that every local summand of M( 0) is a direct summand.
