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Axiomatisability problems for S-systems. (English) Zbl 0637.03029
For a given monoid S (a ring R with a 1), the class of projective S- systems (R-modules) is denoted by P and the class of flat S-systems (R- modules) by F. A monoid or a ring is said to be perfect if $$P=F$$. Necessary and sufficient conditions on a monoid S for the class F to be axiomatizable are given. For example, F is axiomatizable iff every ultrapower of S is flat. G. Sabbagh and P. Eklof [J. Symb. Logic. 36, 623-649 (1971; Zbl 0251.02052)] proved that for a unitary ring R, P is axiomatizable iff F is axiomatizable and R is perfect. It is shown that for a monoid S, if P is axiomatizable, then F is axiomatizable and S satisfies the descending chain condition for principal right ideals. If S satisfies the ascending chain condition for principal left ideals (for example, S is regular), then P is axiomatizable iff F is axiomatizable and S is perfect.