Minimal quasi-ideals of generalized transformation semigroups and generalized rings of linear transformations

Abstract

subsemigroup Q of a semigroup S is called a quasi-ideal of S if SQ [intersection] QS C Q. A quasi-ideal of a ring R is a subring Q of R such that RQ [intersection] QR C Q where RQ [QR] is the set of all finite sums of the form [sigma] r[subscript i] q[subscript i] [[sigma] q[subscript i] r[subscript i]], r[subscript i] Epsilon R and q[subscript i] Epsilon Q. The notion of quasi-ideal was introduced by O. Stienfeld in 1953 and 1956 for rings and semigroups, respectively. By a minimal quasi-ideal of a semigroup S we mean a nonzero quasi-ideal of S which does not properly contain any nonzero quasi-ideal of S. A minimal quasi-ideal of a ring is defined similarly. In 1956, O. Stienfeld characterized minimal quasi-ideals of a semigroup without zero as follows : A quasi-ideal Q of a semigroup S without zero is minimal if and only if Q is a subgroup of S. Also in 1957, he showed that a quasi-ideal of a ring [semigroup with zero] A is either a division subring [subgroup with zero] or a zero subring [zero subsemigroup] of A, and for the first case, the converse holds. Various important semigroups and rings are generalized by using sandwich multiplication. These semigroups and rings are as follows : transformation semigroups, linear transformation semigroups, matrix semigroups, rings of linear transformations and matrix rings. Minimal quasi-ideals of our target generalized semigroups and generalized rings are completely characterized in this research