On monoids over which all strongly flat cyclic right acts are projective. (English) Zbl 0844.20051

Kilp, Mati

Semigroup Forum 52, No. 2, 241-245 (1996).

The author proves that for a monoid \(S\) all strongly flat cyclic right \(S\)-acts are projective if and only if every left collapsible submonoid \(P\) of \(S\) contains a left zero. A submonoid \(P\) is called left collapsible if for any \(p,q\in P\) there exists \(r\in P\) with \(rp=rq\). As a corollary one gets the equivalence of the following statements: (i) All strongly flat cyclic acts are free. (ii) All strongly flat cyclic acts are projective generators in Act-\(S\). (iii) The only left collapsible submonoid of \(S\) is the one-element submonoid.

Reviewer: U.Knauer (Oldenburg)

MSC:

20M50

Connections of semigroups with homological algebra and category theory

Keywords:

projective acts; strongly flat cyclic right \(S\)-acts; left collapsible submonoids; projective generators

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