On semigroups with the idealizer condition (Q1061868)

The following notations are used: I-semigroup - a semigroup in which every proper subsemigroup is distinct from its idealizer; \(I_ 1\)- semigroup - a semigroup having an ascending annihilator series; \(I_ 2\)- semigroup - a semigroup S in which every subsemigroup can be included in a finite ideal series of S; \(I_ 3\)-semigroup - a nilsemigroup in which every proper subsemigroup is nilpotent. The following results are proved: Theorem 1. Every non-trivial commutative I-semigroup has a non-trivial annihilator. Theorem 1 gives a positive answer to Shevrin's problem T 50 b) from ''The Sverdlovsk notebook'' [Semigroup Forum 4, 274-280 (1972; Zbl 0364.20064)]. Corollary 1. Every commutative I-semigroup is an \(I_ 1\)- semigroup. Theorem 2. If an \(I_ 2\)-semigroup S contains no non- nilpotent \(I_ 3\)-subsemigroup, then S is nilpotent. Since every commutative \(I_ 3\)-semigroup is nilpotent (Shevrin, 1961), then it follows Corollary 2. Every commutative \(I_ 2\)-semigroup is nilpotent. Corollary 2 gives a positive answer to Shevrin's problem T 51 b) from ''The Sverdlovsk notebook''.