Sukhendu Kar

DERIVATIONS WITH ANNIHILATOR CONDITIONS IN PRIME RINGS

Abstract. Let Rbe a prime ring, I a nonzero ideal of R, da derivationof R, m(≥ 1),n(≥ 1) two fixed integers and a ∈ R. (i) If a((d(x)y +xd(y) + d(y)x+ yd(x)) n − (xy+ yx)) m = 0 for all x,y ∈ I, then eithera = 0 or R is commutative; (ii) If char(R) 6= 2 and a((d(x)y+ xd(y) +d(y)x+yd(x)) n −(xy+yx)) ∈ Z(R) for all x,y∈ I, then either a= 0 orRis commutative. 1. IntroductionThroughout this paper R always denotes a prime ring with center Z(R),extended centroid C and Q its two-sided Martindale quotient ring. Recall thata ring R is said to be prime, if for any a,b ∈ R, aRb = (0) implies either a = 0or b = 0 and it is semiprime if for any a ∈ R, aRa = (0) implies a = 0. Forany x,y ∈ R, the Lie commutator of x,y is denoted by [x,y] and defined by[x,y] = xy−yx and the anti-commutator is denoted by x◦y and is defined byx ◦ y = xy + yx. By d we mean a derivation of R. A derivation d is inner ifthere exists b ∈ R such that d(x) = [b,x] holds for all x ∈ R.A well known theorem of Posner [15] states that if R is prime and the com-mutator [d(x),x] ∈ Z(R) for all x ∈ R, then either d = 0 or R is commutative.This result of Posner was generalized in many directions by several authorsand they studied the relationship between the structure of prime or semiprimering and the behaviour of additive maps satisfying various conditions. Someauthors have studied the derivations with annihilator conditions in prime andsemiprime rings (see [3], [4], [6], [7], [8], [9]; where further references can befound).In [2], Ashraf and Rehman proved that if R is a prime ring, I is a nonzeroideal of R and d is a derivation of R such that d(x)y+xd(y)+d(y)x+yd(x) =xy + yx for all x,y ∈ I, then R is commutative. Recently, in [1]; Argac andInceboz generalized the above result as follows:

Interval-Valued semiprime fuzzy ideals of semigroups

We introduce the notion of (i-v) semiprime (irreducible) fuzzy ideals of semigroups and investigate its different algebraic properties. We study the interrelation among (i-v) prime fuzzy ideals, (i-v) semiprime fuzzy ideals, and (i-v) irreducible fuzzy ideals and characterize regular semigroups by using these (i-v) fuzzy ideals.

Interval-valued Fuzzy k-ideals and k-regularity of Semirings

In this paper, we characterize k-regularity of semirings and study some important properties of k-regularity of semirings in terms of interval-valued fuzzy k-ideals of semirings.

Characterization of some k-regularities of semirings in terms of fuzzy ideals of semirings

The notion of k-regularity in a semiring generalizes the notion of a regular ring introduced by J. Von Neumaan [15]. In semiring theory, the notion of k-regularity was extensively studied by Sen and Bhuniya [17]. In this paper, we study the concept of k-regularities of a semiring in fuzzy setting and characterize some k-regularities of semirings by using different fuzzy ideals of semirings. We also introduce the concept of power fuzzy semirings and study the notion of k-regularities of power fuzzy semirings with the help of regularities of semigroups.

Soft Ternary Semirings

Abstract In this paper, we introduce the notion of soft ternary semiring by using the concept of soft set theory. Besides, we characterize the notions of regularity and intra-regularity in soft ternary semiring by using different soft (left, lateral, right, quasi, bi) ideals of soft ternary semirings.