The concept of “standard closed ideals” in the weighted discrete abelian semigroup algebra \(l^1(S, \omega )\) is defined in the most general setup. If S is a cancellative, abelian semigroup, then it is shown that the set of all compact elements in \(l^1(S, \omega )\) is always a standard closed ideal. A radical weight \(\omega \) on \({\mathbb {Z}}_+\) is constructed such that \(l^1({\mathbb {Z}}_+, \omega )\) does not have any non-zero compact element. The weighted discrete analogues of some results on the compact elements in \(L^1({\mathbb {R}}_+, \omega )\) given in [2] are proved. Various counter examples are exhibited. Moreover, this article will set the record right by correcting some results of [6].
