The notions of quasi-ideals of rings and semigroups were introduced by Steinfeld (Acta Math. Acad. Sci. Hung. 4:289-298, 1953 and Publ. Math. (Debr.) 4:262-275, 1956) respectively. The notion of -semigroups was introduced by Sen (Proceeding of International Symposium on Algebra and Its Applications, Decker Publication, New York, 1981). Further the notion of (m, n) ideals of semigroups was introduced by Lajos (Acta Sci. Math. 22:217-222, 1961). Later on (m, n) quasi-ideals and (m, n) bi-ideals were widely studied in various algebraic structures viz. semigroups, rings and near-rings etc. In this paper we have defined (m, n) quasi--ideal and (m, n) bi--ideal in -semigroup. Including other results, we have shown that if Q is a minimal (m, n) quasi--ideal in -semigroup S then intersection of minimal m-left -ideals and minimal n-right -ideals is again a minimal (m, n) quasi--ideals.

In this paper, we investigate the commutativity of a semiprime ring R admitting a generalized derivation F with associated derivation D satisfying any one of the properties: (i) F (x) • D(y) = [x, y], (ii) D(x) • F (y) = F [x, y], (iii) D(x) • F (y) = xy, (iv) F (x • y) = [F (x), y] + [D(y), x], and (v) F [x, y] = F (x) • y − D(y) • x for all x, y in some appropriate subsets of R.

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The aim of this paper is to investigate some results of nearrings satisfying certain identities involving generalized derivations. Furthermore, we give some examples to demonstrate the restrictions imposed on the hypothesis of various results which are not superfluous.

There has been considerable interest in the connection between the structure and the -structure of a ring, where denotes an involution on a ring. In this context, Oukhtite and Salhi (2006) introduced a new class or we can say an extension of prime rings in the form of -prime ring and proved several well-known theorems of prime rings for -prime rings. A continuous approach in the direction of -prime rings is still on. In this paper, we establish some results for -prime rings satisfying certain identities involving generalized derivations on -ideals. Finally, we give an example showing that the restrictions imposed on the hypothesis of the various theorems were not superfluous.